# What's the exact definition of the power spectral density function?

I learned from my signal processing course that PSD is the Fourier transform of the autocorrelation function.

$$\mathscr{F}\Big\{\mathrm{E}\big[x(t)x(t+\tau)\big]\Big\}$$

Today I took my statistic course but my professor introduced spectral density function, which is the fourier transform of autocovariance function.

$$\mathscr{F}\Big\{ \mathrm{E}\left[\ \left( x(t)-\mathrm{E}\big[x(t)\big] \right)\left( x(t+\tau)-\mathrm{E}\big[x(t+\tau)\big]\right) \ \right] \Big\}$$

What is the exact definition of PSD?

If it's the FT of autocorrelation, then what's the name for the FT of the autocovariance function?

• Hello, both can be correct and the definition depends on the context, the field, and the information you want to extract from the signal. Check this answer. Oct 31, 2023 at 8:57
• @AlexTP Thanks！ Oct 31, 2023 at 12:44
• I think, because $x(t)$ is usually considered stationary, you need to expect that $$\mathrm{E}\big[x(t+\tau)\big]=\mathrm{E}\big[x(t)\big]$$ and all other expectation values are independent of any translation in time by $\tau$ or any other offset. Oct 31, 2023 at 16:11
• @robertbristow-johnson Thank you! Nov 3, 2023 at 10:29

The problem is that the terms autocorrelation and autocovariance are sometimes used to mean different things. In statistics, they often use autocorrelation for what would be called autocovariance in the signal processing literature (cf. this post over at stats.SP).

The power spectrum is the Fourier transform of what in the signal processing literature is called autocorrelation function:

$$S_x(\omega)=\int_{-\infty}^{\infty}R_x(\tau)e^{-j\omega\tau}d\tau$$

with

$$R_x(\tau)=E\big\{x(t)x(t+\tau)\big\}$$

where I've assumed that $$x(t)$$ is a wide-sense stationary (WSS) random process.

The Fourier transform of the autocovariance

$$C_x(\tau)=E\big\{[x(t)-\mu_x][x(t+\tau)-\mu_x]\big\}=R_x(\tau)-\mu_x^2$$

is sometimes used for processes with a non-zero mean $$\mu_x$$ to avoid the Dirac delta impulse in the power spectrum at $$\omega=0$$. The Fourier transform of the autocovariance is called the covariance spectrum of $$x(t)$$.