Fourier transform of a signal and its autocovariance function

Question

While I do know the difference between the two, in theory, I am not very sure about why we look at the Fourier transform of the auto-covariance function. What extra information does it give us over and above what we can already get by taking the Fourier transform of the original time-series? Also, when is one advantageous over the other?

Answer 1

The interest in looking at the Fourier transform of the autocorrelation (which is usually equivalent to autocoveriance) comes from the Wiener-Khinchin theorem that states

Consider a wide-sense stationary signal $x(t)$. Then the power spectral density $S_x(f)$ is given by

$$S_x(f) = \int_{-\infty}^{\infty}R(\tau)e^{-j2{\pi}f{\tau}}d{\tau}$$

Where $R(\tau)$ is the autocorrelation function of $x(t)$.

There are times where the Fourier transform of a signal does not exist. However, this result still allows us to analyze the signal in the frequency domain and retrieve information by using the autocorrelation and not the signal itself.

This is also helpful when analyzing LTI systems. Given a system $h(t)$ with input $x(t)$ and output $y(t)$, we have a relationship between the power spectral densities given by

$$S_y(f) = S_x|H(f)|^2$$

This aids in estimating properties of the signals as well as the system itself, depending on what information is already known.

In the discrete domain, the DFT can always be calculated for a finite-length signal. The theorem still holds however, and despite being able to "take" the Fourier transform as in the DFT, errors may begin to appear when estimating parameters.

References by OP's request:

https://en.wikipedia.org/wiki/Wiener%E2%80%93Khinchin_theorem

https://mathworld.wolfram.com/Wiener-KhinchinTheorem.html

https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-011-introduction-to-communication-control-and-signal-processing-spring-2010/readings/MIT6_011S10_chap10.pdf