# when to use auto correlation and when to use fourier transform

In communication, if we don't know any prior information about the signal, we perform an auto correlation say Rx(T) and take the Fourier transform of Rx(T) to analyze the power spectrum.

Supposing if my signal is known do i need to perform auto correlation , or simply i can take a Fourier transform, and then calculate the power from it.?

Am I seeing it right? or am I missing something here.?

thanks in advance phani tej

• If you don't find an adequate answer here, you might also try DSP. A similar question might even already have an answer over there. – The Photon Oct 22 '14 at 16:18

## 1 Answer

It all boils down to the question whether the Fourier transform of the signal exists or not. If you model a signal as a stochastic process then you can't take its Fourier transform because normally it doesn't exist. But if you consider a real-world signal (which usually has finite length) then the Fourier transform (usually) exists and the two operations (autocorrelation + Fourier transform, Fourier transform + magnitude squared) are identical. In that case the (deterministic) autocorrelation is

$$R_x(\tau)=\int_{-\infty}^{\infty}x(t)x(t-\tau)dt=x(t)*x(-t)$$

where $*$ denotes convolution. Noting that the Fourier transform of $x(-t)$ is $X(-\omega)=X^*(\omega)$ (because $x(t)$ is real-valued), you get for the Fourier transform of the autocorrelation

$$\mathcal{F}\{R_x(\tau)\}=X(\omega)X^*(\omega)=|X(\omega)|^2$$

So either computing the autocorrelation and taking its Fourier transform or computing the signal's Fourier transform and taking the squared magnitude give the same result. Note that this is true whenever the Fourier transform of the signal exists.