Wiener-Khinchin Theorem

About

There is an interesting theorem from long ago that often appears in the fields of stochastic processes and signal processing: the Wiener-Khinchin theorem.
It is a very simple theorem, stating that the Fourier transform of the autocorrelation function is equal to the power spectrum.

The theorem

The autocorrelation function can be expressed as follows:

R_X(\tau) = \mathbb{E}[X(t) X(t+\tau)]

Here:

• R_X(\tau) is the autocorrelation function.
• \mathbb{E}[\cdot] denotes the expected value.
• X(t)X(t) is the stochastic process.
• \tau represents the time lag.

Thus, as mentioned above, this Fourier transform corresponds to the power spectrum.

S_X(f) = \int_{-\infty}^\infty R_X(\tau) e^{-j 2 \pi f \tau} \, d\tau

But isn’t the power spectrum just obtained by simply taking the Fourier transform of the signal and then taking its absolute value? What’s the difference?

S_X(f) = \left| \int_{-\infty}^\infty X(t) e^{-j 2 \pi f t} \, dt \right|^2

The difference

The difference lies in whether the power spectrum determined from the auto-correlation function is averaged or not. The spectrum derived from the auto-correlation function represents an average power spectrum, while the one obtained by directly taking the Fourier transform of the signal and its magnitude reflects an instantaneous power spectrum.

In other words, while the power spectrum is obtained by performing the Fourier transform over the entire data range, the autocorrelation function, by its nature, requires correlation values, which means it needs the average correlation over a specific data segment. For this reason, it is not possible to perform a Fourier transform over the entire data range, resulting in an averaged spectrum. Consequently, the frequency resolution decreases. However, a stable spectrum can be obtained. Another interesting point is that if the autocorrelation function is known, it is possible to calculate the spectrum for a specific frequency range.