Power Spectral Density

The same stationary random process seen as time-domain correlation and frequency-domain power, linked by the Wiener-Khinchin theorem.

This page is the representation view: autocorrelation and spectra. For the system view, where a filter acts on random inputs in both time and frequency, see LTI Systems on Random Inputs.

1. Wide-sense stationarity

A random process $X(t)$ is wide-sense stationary when its mean is constant and its autocorrelation depends only on lag:

$$ R_X(\tau)=\mathbb{E}[X(t)X(t+\tau)]. $$

That lag-only dependence is what makes a spectrum meaningful. The process may wiggle differently on each realization, but its average second-order structure is fixed in time.

WSS vs SSS. Strict-sense stationarity (SSS) is the stronger condition that the full joint distribution of $(X(t_1),\dots,X(t_n))$ is invariant under any time shift. Wide-sense stationarity (WSS) only constrains the first two moments: constant mean and lag-only autocorrelation. SSS implies WSS, but not the reverse. The spectral story on this page is exactly the WSS story: Wiener-Khinchin needs nothing beyond second-order structure. For a Gaussian process the two coincide, because a Gaussian is fully determined by its mean and autocorrelation.

Figure 0 · Wide-sense stationarity as stable second-order structure

W.S.S. process non-W.S.S. process running summaries

PSD in one line. The power spectral density says how the total second-moment power of a W.S.S. process is distributed over frequency.

2. Wiener-Khinchin as a Fourier pair

The Wiener-Khinchin theorem says the autocorrelation and power spectral density are Fourier transforms of one another:

$$ S_X(\omega)=\int_{-\infty}^{\infty} R_X(\tau)e^{-i\omega\tau}\,d\tau,\qquad R_X(\tau)=\frac{1}{2\pi}\int_{-\infty}^{\infty} S_X(\omega)e^{i\omega\tau}\,d\omega. $$

For a real W.S.S. process, $S_X(\omega)$ is real, even, and nonnegative. The area under it is total power:

$$ R_X(0)=\mathbb{E}[X(t)^2]=\frac{1}{2\pi}\int_{-\infty}^{\infty}S_X(\omega)\,d\omega. $$

Figure 1 · Autocorrelation and PSD as a Fourier pair

$R(\tau)$ $S(\omega)$

band / color 0.55

center frequency 0

3. Sample paths and audio texture

The PSD is not a single waveform; it describes the average frequency content of a population of waveforms. Regenerating sample paths from the same spectrum makes this visible: the paths differ, but their roughness, dominant oscillations, and slow drift match the same spectral shape.

Figure 2 · Fresh sample paths from the current spectrum

Figure 2b · Spectrogram: time-local frequency power

sample path local high power instantaneous tone

chirp strength 0.65

window length 56

4. LTI shaping

A linear time-invariant filter shapes spectra by multiplication:

$$ S_{YY}(\omega)=|H(\omega)|^2S_{XX}(\omega). $$

The filter ignores the random phases of the input; at the second-order level it scales each frequency's power.

Figure 3 · Input PSD, filter gain, and output PSD

$S_{XX}$ $|H|^2$ $S_{YY}$

filter center 0

filter bandwidth 1

5. Periodogram estimates

The PSD is an ensemble object, but real data gives you finite records. The periodogram estimates power by squaring Fourier coefficients from one sample path. It is noisy: longer records reveal the spectral shape better, but single-frequency estimates still fluctuate.

Figure 4 · Periodogram convergence with record length

Figure 4b · Welch averaging and window choice

true PSD raw periodogram Welch estimate

segments 6

window

What to remember

Object	Time-domain reading	Frequency-domain reading
$R(\tau)$	How much a process resembles a lagged copy of itself.	Narrow spectra produce long-lived oscillatory correlations.
$S(\omega)$	Determines the autocorrelation by inverse Fourier transform.	How power is distributed over frequency; real, even, nonnegative for real W.S.S. processes.
LTI filter	Convolution reshapes sample paths.	Multiplication by $\|H(\omega)\|^2$ reshapes PSDs.
Periodogram	Computed from one finite realization.	A noisy estimate of the underlying PSD.

What next

PSD sits between Fourier analysis, filtering, and stochastic-process inference.

Systems

LTI Systems on Random Inputs

Use the PSD to predict how linear filters reshape random sample paths, correlations, and spectra.

Inference

Monte Carlo & Bayesian Sampling

Compare frequency-domain second-order structure with simulation methods for posterior distributions.

Information

Entropy & Mutual Information

Move from second-order power to uncertainty and dependence between random variables.

Geometry

Fisher Information

Another way expectations turn local model behavior into a global quantitative object.

Regression

Gaussian Processes

Use kernels as covariance functions for priors over functions, then condition them on observed data.