PowerSpectralDensity

In a wide-sense stationary (WSS) stochastic process, certain statistical properties, such as mean and variance, are constant over time, and the covariance between two time points depends only on the distance between those points and not on the actual time at which the covariance is computed.

The power spectral density (PSD) is a very useful concept in the analysis of stochastic processes. It can be thought of as providing the frequency content of a stochastic process, and is related to the autocorrelation of the process by the Wiener-Khinchin theorem, which states that the autocorrelation function and the PSD are Fourier transform pairs.

In the context of a continuous-time WSS stochastic process, for example, if $X(t)$ is our process, then the PSD $S_X (f)$ would be defined as the Fourier transform of the autocorrelation function $R_X (\tau)$ of $X (t)$:

$$S_X (f) = \int R_X (\tau) e^{- 2 \pi f \tau} d \tau$$

The PSD provides a measure of the power or strength at each frequency $f$ that the process contains. Here, power is measured by the average squared amplitude of the process at frequency $f$.

This mathematical abstraction is quite powerful and general, and can be applied to different types of stochastic processes beyond just time series. For example, it can be applied to spatial processes (like a field of noise over a 2D space), or to more abstract spaces and random processes defined on them.

The PSD provides a measure of the process in the frequency domain and is a key tool in theoretical and applied investigations of stochastic processes. The techniques used to calculate and interpret the PSD in these more abstract settings can be quite sophisticated and depend on the specific characteristics of the process and its domain.