Spectral Density

The power spectral density (PSD) describes how the power (or variance) of a time series is distributed with frequency. Mathematically, it is defined as the Fourier Transform (FT) of the autocorrelation sequence of the time series.

A spectral density function of a stochastic (random) process is a Fourier transform of its covariance function

.

In the same way, the cospectral density function of the random processes and can be evaluated as follows

; ,

where is Fourier operator.

and are closely related. Namely, the wider plot of then narrower plot for (it means that a random variable changes with time slowly) and vice versa, the narrower plot of then the wider plot for (in this case, a random variable changes fast with time).

White Noise

White noise is a collection of uncorrelated random variables with constant mean and variance. The spectral density of the white noise does not depend on frequency and possesses with a constant value, i.e. . In practice, physical systems are never disturbed by white noise, although white noise is a useful theoretical approximation when the noise disturbance has a correlation time that is very small relative to the natural bandwidth of the system.

The white noise correlation function and its specral density are given in Fig.2.

a) b)

Fig.2 White noise correlation function a) and its power spectral density function b)

Дата добавления: 2015-11-14