PowerSpectrum

The power spectrum is a common tool used in the analysis of stochastic processes, and is defined as the Fourier transform of the autocorrelation function.

Let's define a stochastic process ${X(t), t \in \mathbb{R}}$. The autocorrelation function $R(\tau)$ of the stochastic process $X(t)$ is defined as:

$$R(\tau) = E[(X(t) - \mu)(X(t+\tau) - \mu)]$$

where $E[.]$ denotes the expected value, $\mu$ is the mean of the process, and $\tau$ is the time lag.

The power spectrum $S(f)$ of this process is then the Fourier transform of the autocorrelation function:

$$S(f) = \int R(\tau) e^{-2\pi i f \tau} d\tau$$

where $f$ is frequency and the integral is taken over all $\tau$. This gives the distribution of power as a function of frequency. It can be understood as the decomposition of the process's variance across different frequency components.

An important property of the power spectrum is that its total power (obtained by integrating the power spectrum over all frequencies) is equal to the variance of the process:

$$\int S(f) df = \text{Var}[X(t)]$$

where $\text{Var}[.]$ denotes the variance. This property follows from the Wiener-Khinchin theorem, which links the power spectrum and the autocorrelation function in this way.

Note: The power spectrum is generally a function of frequency for continuous-time processes and a function of wavenumber for discrete-time processes. The actual calculations can get quite involved depending on the specific properties of the stochastic process under consideration. The equations here are presented for the simplest case of a stationary (i.e., statistical properties do not change over time) stochastic process with zero mean. The power spectrum for non-stationary processes or processes with non-zero mean would need additional considerations.