Power spectral density (part 2)

Last time we have written on the power spectral density and we have "analyzed" deterministic periodic time series. This time we will consider spectral densities of some stochastic processes.

Spectral density of the white noise

As we have already written before, the white noise is a purely random, non-correlated (possessing no memory), noise. For the white noise the following is true:

\begin{equation} \left\langle \xi(t_1) \xi(t_2) \right\rangle =\delta(t_1 - t_2) , \end{equation}

here \( \xi \) is a white noise (formally a function of time), chevrons denotes averaging over different time moments, while \( \delta \) is a Dirac delta function.

As such signal has no persistent trends, its spectral density is always related to the standard deviation, \( \sigma \), of the underlying white noise:

\begin{equation} S(\nu) = \sigma^2 . \end{equation}

There is no dependence on the frequency, because there is no persistent trends or alternatively - all trends are reflected equally in the observed time series.

Spectral density of the Brownian motion

Brownian motion, also known as Wiener process or "brown" noise, is a motion affected by many random perturbations. One of the most well known examples for such motions is a movements of pollen immersed in water. The pollen is affected by many hits incoming from the water molecules, therefore it moves in seemingly random pattern. Mathematically Brownian motion is defined as a sum, or integral, of the white noise:

\begin{equation} \mathrm{d} W(t) = W(t+\mathrm{d} t) - W(t) = \xi(t)\mathrm{d} t , \end{equation}

\begin{equation} W(t) = \int\limits_0^t \mathrm{d} W(t) , \quad W_t =\sum_{i=0}^t \xi_i \Delta_i . \end{equation}

The mean of the time series remains zero, but the variance is proportional to the time. The spectral density of such time series is no longer constant:

\begin{equation} S(\nu) = \frac{\sigma^2}{2 \pi^2 \nu^2} . \end{equation}

Spectral density of the Brownian motion with relaxation

In most of the real physical situations friction is present, therefore it proves useful to extend the Brownian motion by implementing some kind of relaxation term:

\begin{equation} \mathrm{d} f(t) = - \gamma f(t) \mathrm{d} t + \sigma\mathrm{d} W(t) . \end{equation}

In this spectral density becomes slightly more complex:

\begin{equation} S(\nu) = \frac{\sigma^2}{2 \gamma^2 + 2 \pi^2 \nu^2} . \end{equation}

The mathematical form of spectral density suggests that in certain range of frequencies, \( \nu \ll \gamma \), white noise will prevail, while for higher frequencies brown noise will be observed.

Geometric Brownian motion

Another interesting take on the Brownian motion is known as geometric Brownian motion. Generally speaking it is mostly the same as Brownian motion, but it is not additive, but multiplicative! Namely Brownian motion occurs on the logarithmic scale. The stochastic differential equation for the simplified geometric Brownian motion can be mathematically expressed as follows:

\begin{equation} \mathrm{d} f(t) = \sigma f(t) \mathrm{d} W(t) . \end{equation}

Spectral densities of the aforementioned stochastic processes

Below you should see an interactive program, which enables you to analyze spectral densities of the stochastic processes discussed above. \( \sigma \) slider is important at all times as it controls the standard deviation of the underlying (the time series of the other stochastic processes are derived from it) white noise. \( \gamma \) plays a role only if the Brownian motion is plotted. Most of the spectral density plots are approximated by a red curves, which stand for the theoretical expressions of spectral density provided in the text above.

Previously the interactive app was powered by Wolfram CDF technology, but it is now replaced with HTML5 app. The old app can still be downloaded from here.