Power spectral density of stationary processes

The Power Spectral Density is defined as the Fourier transform of the autocorrelation function. It is only defined for Stationary Processes. It provides information about the distribution of power in the frequency domain:

S_{X X} (Ω) = \int_{- \infty}^{\infty} R_{X X} (τ) e^{- j Ω τ} d τ \leftrightarrow R_{X X} (τ) = \frac{1}{2 π} \int_{- \infty}^{\infty} S_{X X} (Ω) e^{j Ω τ} d Ω

S_{X X} (ω) = \sum_{n = - 1}^{N} R_{X X} (m) e^{- j ω m} \leftrightarrow R_{X X} [m] = \frac{1}{2 π} \int_{- π}^{π} S_{X X} (ω) e^{j ω m} d ω

Properties:

The Power Spectral Density is a non-negative function.
The random process power is obtained by integrating the power spectral density:

$P_{x} = R_{X X} (0) = \frac{1}{2 π} \int_{- \infty}^{\infty} S_{X X} (Ω) d Ω$ (for continuous time processes)

$P_{x} = R_{X X} (0) = \frac{1}{2 π} \int_{- π}^{π} S_{X X} (ω) d ω$ (for discrete time processes)
The Power Spectral Density is an even function If the process is real: $S_{X X} (ω) = S_{X X} (- ω)$

Review of Basic Concepts of Probability

Power spectral density of stationary processes

Properties: