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)