White noise is not a specific signal but a statistical model that describes a random signal having equal intensity at different frequencies, giving it a constant power spectral density. This theoretical construct draws its name from white light, although light that appears white generally does not have a flat power spectral density over the visible band. In the physical world, an ideal white noise signal is a purely theoretical construction because the bandwidth is limited in practice by the mechanism of noise generation, by the transmission medium, and by finite observation capabilities. For an audio signal, the relevant range is the band of audible sound frequencies between 20 and 20,000 Hz, and such a signal is heard by the human ear as a hissing sound resembling the /h/ sound in a sustained aspiration. The concept can be defined for signals spread over more complicated domains, such as a sphere or a torus, and in discrete time, white noise is a discrete signal whose samples are regarded as a sequence of serially uncorrelated random variables with a mean of zero and a finite variance. A single realization of white noise is a random shock, and in some contexts, it is also required that the samples be independent and have identical probability distribution. In digital image processing, the pixels of a white noise image are typically arranged in a rectangular grid and are assumed to be independent random variables with uniform probability distribution over some interval. The term is used with this or similar meanings in many scientific and technical disciplines, including physics, acoustical engineering, telecommunications, and statistical forecasting.
The Physics of Randomness
Any distribution of values is possible for white noise, although it must have zero DC component, and even a binary signal which can only take on the values 1 or -1 will be white if the sequence is statistically uncorrelated. It is often incorrectly assumed that Gaussian noise necessarily refers to white noise, yet neither property implies the other, as Gaussianity refers to the probability distribution with respect to the value while the term white refers to the way the signal power is distributed independently over time or among frequencies. One form of white noise is the generalized mean-square derivative of the Wiener process or Brownian motion, and a generalization to random elements on infinite dimensional spaces, such as random fields, is the white noise measure. In the mathematical field known as white noise analysis, a Gaussian white noise is defined as a stochastic tempered distribution, which is a random variable with values in the space of tempered distributions. A probability law on the infinite-dimensional space can be defined via its characteristic function, and the white noise must satisfy specific conditions involving the natural pairing of the tempered distribution with the Schwartz function. However, a precise definition of these concepts is not trivial because some quantities that are finite sums in the finite discrete case must be replaced by integrals that may not converge. Indeed, the set of all possible instances of a signal is no longer a finite-dimensional space but an infinite-dimensional function space, and by any definition a white noise signal would have to be essentially discontinuous at every point. Therefore, even the simplest operations on the signal, like integration over a finite interval, require advanced mathematical machinery, and most authors define the signal indirectly by specifying random values for the integrals of the signal over each interval.