# Clarification concerning power spectral density

Many books in signal processing, e.g. Papoulis [1], define power spectral density (PSD) as:

$$S(\omega)=\sum_{k=-\infty}^{\infty}R_{xx}(k)e^{-j\omega k}$$

Which is the fourier transform of the correlation function: $$R_{xx}(\tau)=E[x(t)x(t-\tau)]$$

However, some authors in the context of time series analysis, for example Jenkins [2] define it as:

$$\Gamma_{xx}(\omega)=\sum_{k=-\infty}^{\infty}\gamma_{xx}(k)e^{-j\omega k}$$

Which is the fourier transform of the covariance function:

$$\gamma_{xx}(\tau)=\hbox{Cov}[x(t)x(t-\tau)]=E[x(t)x(t-\tau)]-E[x(t)]E[x(t-\tau)]$$

As I understand, the Wiener-Khinchin theorem considers $R_{xx}(\tau)$

Can someone clarify why $\Gamma_{xx}(\omega)$ is a valid definition of PSD.

[1] Papoulis, A. (1965). Probability, random variables, and stochastic processes.

[2] Jenkins,G. Watts,D. (1968), Spectral Analysis and Its Applications

I haven't seen Jenkins' context but for zero mean WSS random processes the autocorrelation function and the auto covariance function will the the same, hence you can use either of the definitions when the mean is zero.

For nonzero mean processes, see the relation between autocorrelation and autocovariance functions in p.321 of Papoulis 3ed. (eq. 10.125) , which is $$S(\omega) = S^c(\omega) + 2\pi \eta^2\delta(\omega)$$

Where $S(\omega)$ is the aurocorrelation function and $S^c(\omega)$ is the autocovariance function and $\eta$ is the mean of the WSS random process.

Note that the autocovariance definition will always treat the random process by subtracting its mean.

Considering you are using PSD and not BiSpectra implicitly impose you assumed your process is WSS (you only consider the delay between samples and not their absolute time regarding the time origin).

Considering WSS assumption your process' ensemble mean is constant and doesn't change with time (E[x(t)]=Constant). So the only difference between co-variance and auto-correlation is a DC bias which only affect your DC component of your PSD so for zero mean processes(E[x(t)]=0) there won't be any difference between these definitions.

Considering practical estimation of PSD always there is a large DC component in PSD which always leaks to other frequencies and mask them out. So in practice they usually remove the DC component (removes a DC bias from the process then calculate the PSD).