White noise
A generalized stationary stochastic process
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) |
White noise. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=White_noise&oldid=49205