White noise

A generalized stationary stochastic process $X( t)$ with constant spectral density. The generalized correlation function of white noise has the form $B( t) = \sigma ^ {2} \delta ( t)$, where $\sigma ^ {2}$ is a positive constant and $\delta ( t)$ is the delta-function. The white noise process is extensively applied in describing random disturbances with a very small correlation period (e.g. "thermal noisethermal noise" — pulsations of the current intensity in a conductor, generated by the thermal motion of the electrons). In the spectral decomposition of white noise,

$$X ( t) = \int\limits _ {- \infty } ^ \infty e ^ {i \lambda t } dz ( \lambda ),$$

the "elementary vibrations" $e ^ {i \lambda t } dz ( \lambda )$ have, on the average, the same intensity at all frequencies $\lambda$; more accurately, their average squared amplitude is

$${\mathsf E} | dz ( \lambda ) | ^ {2} = \ \frac{\sigma ^ {2} }{2 \pi } d \lambda ,\ \ - \infty < \lambda < \infty .$$

This spectral decomposition means that, for each square-integrable function $\phi ( t)$,

$$\langle X , \phi \rangle = \ \int\limits _ {- \infty } ^ \infty \phi ( t) X ( t) dt = \ \int\limits _ {- \infty } ^ \infty \widetilde \phi ( \lambda ) dz ( \lambda ),$$

where $\widetilde \phi ( \lambda )$ is the Fourier transform of $\phi ( t)$; a more explicit dependence of the generalized process $X = \langle X, \phi \rangle$ on the function $\phi ( t)$ may be described by a corresponding stochastic measure $d \eta ( t)$ of the same type as $dz( \lambda )$( $d \eta ( t)$ is the Fourier transform of the stochastic measure $dz ( \lambda )$), viz.

$$\langle X , \phi \rangle = \int\limits _ {- \infty } ^ \infty \phi ( t) d \eta ( t).$$

Gaussian white noise $X( t)$, which is the generalized derivative of Brownian motion $\eta ( t)$( $X( t) = \eta ^ \prime ( t)$), is the basis for constructing stochastic diffusion processes $Y( t)$( cf. Diffusion process), "controllable" by a stochastic differential equation:

$$Y ^ \prime ( t) = a ( t, Y ( t)) + \sigma ( t, Y ( t)) \cdot \eta ^ \prime ( t).$$

These equations are often written in the form of differentials:

$$dY ( t) = a ( t, Y ( t)) dt + \sigma ( t, Y ( t)) d \eta ( t).$$

Another important model involving the use of white noise is the stochastic process $Y( t)$ which describes the behaviour of a stable vibrating system acted upon by stationary random perturbations $X( t)$, when $Y( s)$, $s < t$, do not depend on $X( u)$, $u > t$. A very simple example of this is the system

$$P \left ( \frac{d}{dt} \right ) Y ( t) = X ( t),$$

where $P( z)$ is a polynomial with roots in the left half-plane; after the damping of the "transient processes" , the process $Y( t)$ is given by

$$Y ( t) = \int\limits \frac{1}{P ( i \lambda ) } dz ( \lambda ).$$

In practical applications, in the description of the so-called shot effect process, white noise of the form

$$X ( t) = \sum _ { k } \delta ( t - \tau _ {k} )$$

plays an important role ( $k$ varies between $- \infty$ and $\infty$ and the $\dots \tau _ {-} 1 , \tau _ {0} , \tau _ {1} \dots$ form a Poisson process); more accurately, $X( t)$ is the generalized derivative of a Poisson process $\eta ( t)$. The shot effect process itself has the form

$$Y ( t) = \ \int\limits _ {- \infty } ^ \infty c ( t, s) X ( s) ds = \ \int\limits _ {- \infty } ^ \infty c ( t, s) d \eta ( s ) =$$

$$= \ \sum _ { k } c ( t, \tau _ {k} ),$$

where $c( t, s)$ is some weight function satisfying the condition

$$\int\limits _ {- \infty } ^ \infty | c( t, s ) | ^ {2} ds < \infty ;$$

in addition, the average value of the generalized process $X = \langle X, \phi \rangle$ is

$$a ( \phi ) = a \int\limits _ {- \infty } ^ \infty \phi ( t) dt.$$

Here, $a$ is the parameter of the Poisson law (see above), and the stochastic measure $dz ( \lambda )$ in the spectral representation

$$X ( t) = a + \int\limits _ {- \infty } ^ \infty e ^ {i \lambda t } dz ( \lambda )$$

of this process is such that

$${\mathsf E} | dz ( \lambda ) | ^ {2} = \frac{a}{2 \pi } d \lambda .$$

References

[1]	Yu.V. [Yu.V. Prokhorov] Prohorov, Yu.A. Rozanov, "Probability theory, basic concepts. Limit theorems, random processes" , Springer (1969) (Translated from Russian)

Comments

See [a1] for applications of white noise as the limit of "wide bandwidth" noise in physical systems and [a2] for the relationship between differential equations with white noise inputs and the stochastic differential equations of Itô calculus (cf. also Itô formula; Stochastic differential equation). See also Stratonovich integral for further information on this topic. Further important topics are the analysis of white noise regarded as a generalized random function [a3], i.e. a probability on the space ${\mathcal S} ^ \prime$ of tempered distributions on $[ 0, \infty )$( cf. White noise analysis), and application of white noise theory in non-linear filtering [a4], where "white noise" is interpreted in terms of finitely-additive Gaussian measures on cylinder sets of a separable Hilbert space.

References

[a1]	H.J. Kushner, "Approximation and weak convergence methods for random processes, with applications to stochastic systems theory" , M.I.T. (1984)
[a2]	N. Ikeda, S. Watanabe, "Stochastic differential equations and diffusion processes" , North-Holland & Kodansha (1988)
[a3]	T. Hida, "Brownian motion" , Springer (1980)
[a4]	G. Kallianpur, R.L. Karandikar, "White noise theory of prediction, filtering and smoothing" , Gordon & Breach (1988)
[a5]	I.M. Gel'fand, N.Ya. Vilenkin, "Generalized functions. Applications of harmonic analysis" , 4 , Acad. Press (1968) pp. Chapt. III (Translated from Russian)
[a6]	T. Hida (ed.) H.-H. Kuo (ed.) J. Potthoff (ed.) L. Streid (ed.) , White noise analysis - mathematics and applications , World Sci. (1990)