White Noise

White noise () is a random Signal (or process) with a flat Power Spectral Density . In other words, the signal's power spectral density has equal power in any band, at any centre frequency, having a given bandwidth. White noise is so called as an analogy with white light which contains all frequencies.

An infinite-bandwidth white noise signal is purely a theoretical construct. By having power at all frequencies, the total power of such a signal is infinite. In practice, a signal can be "white" with a flat spectrum over a defined frequency band.

STATISTICAL PROPERTIES

The term white noise is also commonly applied to a noise signal in the spatial domain which has zero Autocorrelation over the relevant space dimensions. The signal is then "white" in the Spatial Frequency domain (this is equally true for signals in the angular frequency domain, e.g. the distribution of a signal across all angles in the night sky). The image to the right displays a finite length, discrete time realization of a white noise process generated from a computer.

Being uncorrelated in time does not, however, restrict the values a signal can take. Any distribution of values is possible (although it must have zero DC component). For example, a binary signal which can only take on the values 1 or 0 will be white if the sequence of zeros and ones is statistically uncorrelated. Noise having a continuous distribution, such as a Normal Distribution , can of course be white.

It is often incorrectly assumed that Gaussian Noise (i.e. noise with a Gaussian amplitude distribution - see Normal Distribution ) is necessarily white noise. However, neither property implies the other. Gaussianity refers to the way signal values are distributed, while the term 'white' refers to correlations at two distinct times, which are independent of the noise amplitude distribution.

We can therefore find Gaussian white noise, but also Poisson, Cauchy, etc. white noises. Note that the distribution must have infinite variance. Thus, the two words "Gaussian" and "white" are often both specified in mathematical models of systems. Gaussian white noise is a good approximation of many real-world situations and generates mathematically tractable models. These models are used so frequently that the term . Gaussian white noise has the useful statistical property that its values are independent (see Statistical Independence ).

White noise is the generalized mean-square derivative of the Wiener Process or Brownian Motion .

COLORS OF NOISE

See Also: Colors of noise

There are also other "colors" of noise, the most commonly used being Pink and Brown .

APPLICATIONS

One use for white noise is in the field of Architectural Acoustics . Here in order to submerge distracting, undesirable noises (for example conversations, etc.) in interior spaces, a constant low level of noise is generated and provided as a background sound. White noise is used by some Siren s for emergency vehicles, due to its ability to cut through background noise (e.g. urban traffic noise).

White noise has also been used in Electronic Music , where it is used either directly or as an input for a filter to create other types of noise signal. It is used extensively in Audio Synthesis , typically to recreate percussive instruments such as Cymbals which have high noise content in their frequency domain.

It is also used to generate Impulse Responses . To set up the EQ for a concert or other performance in a venue, a short burst of pink noise is sent through the PA system and monitored from various points in the venue so that the engineer can tell if the acoustics of the building naturally boost or cut any frequencies. He or she can then adjust the overall EQ to ensure a balanced mix.

White noise is used as the basis of some Random Number Generators .

White noise can be used to disorient individuals prior to Interrogation and may be used as part of Sensory Deprivation techniques. White Noise Machine s are sold as privacy enhancers and sleep aids.

MATHEMATICAL DEFINITION

White random vector

A random vector

\mathbf{w}

is a white random vector if and only if its Mean Vector and Autocorrelation Matrix are the following:
:

\mu_w =  \mathbb{E}\{ \mathbf{w} \} = 0

R_{ww} = \mathbb{E}\{ \mathbf{w} \mathbf{w}^T\} = \sigma^2 \mathbf{I}

I.e., it is a zero mean random vector, and its autocorrelation matrix is a multiple of the Identity Matrix . When the autocorrelation matrix is a multiple of the identity, we say that it has spherical correlation.

White random process (white noise)

A continuous time random process

w(t)

where

t \in \mathbb{R}

is a white noise process if and only if its mean function and autocorrelation function satisfy the following:
:

\mu_w(t) =  \mathbb{E}\{ w(t)\} = 0

R_{ww}(t_1, t_2) = \mathbb{E}\{ w(t_1) w(t_2)\} = \sigma^2 \delta(t_1 - t_2)

I.e., it is a zero mean process for all time and has infinite power at zero time shift since its autocorrelation function is the Dirac Delta Function .

The above autocorrelation function implies the following power spectral density.
:

S_{xx}(\omega) = \sigma^2 \,\!

since the Fourier Transform of the Delta Function is equal to 1. Since this Power Spectral Density is the same at all frequencies, we call it ''white'' as an analogy to the Frequency Spectrum of White Light .

RANDOM VECTOR TRANSFORMATIONS

Two theoretical applications using a white random vector are the ''simulation'' and ''whitening'' of another arbitrary random vector. To ''simulate'' an arbitrary random vector, we transform a white random vector with a carefully chosen matrix. We choose the transformation matrix so that the mean and Covariance Matrix of the transformed white random vector matches the mean and Covariance Matrix of the arbitrary random vector that we are simulating. To ''whiten'' an arbitrary random vector, we transform it by a different carefully chosen matrix so that the output random vector is a white random vector.

These two ideas are crucial in applications such as Channel Estimation and Channel Equalization in Communications and Audio . These concepts are also used in Data Compression .

Simulating a random vector

Suppose that a random vector

\mathbf{x}

has Covariance Matrix

K_{xx}

. Since this matrix is Hermitian Symmetric and Positive Semidefinite , by the Spectral Theorem from Linear Algebra , we can diagonalize or factor the matrix in the following way.
:

\,\! K_{xx} = E \Lambda E^T

where

E

is the Orthogonal Matrix of Eigenvector s and

\Lambda

is the Diagonal Matrix of Eigenvalue s.

We can simulate the 1st and 2nd Moment properties of this Random Vector

\mathbf{x}

with Mean

\mathbf{\mu}

and covariance matrix

K_{xx}

via the following transformation of a white vector

\mathbf{w}

:
:

\mathbf{x} = H \, \mathbf{w} + \mu

where
:

\,\!H = E \Lambda^{1/2}

Thus, the output of this transformation has expectation
:

\mathbb{E} \{\mathbf{x}\} = H \, \mathbb{E} \{\mathbf{w}\} + \mu = \mu

and covariance matrix
:

\mathbb{E} \{(\mathbf{x} - \mu) (\mathbf{x} - \mu)^T\} = H \, \mathbb{E} \{\mathbf{w} \mathbf{w}^T\} \, H^T = H \, H^T = E \Lambda^{1/2} \Lambda^{1/2} E^T = K_{xx}

Whitening a random vector

The method for whitening a vector

\mathbf{x}

with Mean

\mathbf{\mu}

and Covariance Matrix

K_{xx}

is to perform the following calculation:
:

\mathbf{w} = \Lambda^{-1/2}\,  E^T \, ( \mathbf{x} - \mathbf{\mu} )

Thus, the output of this transformation has expectation
:

\mathbb{E} \{\mathbf{w}\} = \Lambda^{-1/2}\,  E^T \, ( \mathbb{E} \{\mathbf{x} \} - \mathbf{\mu} ) = \Lambda^{-1/2}\,  E^T \, (\mu - \mu) = 0

and covariance matrix
:

\mathbb{E} \{\mathbf{w} \mathbf{w}^T\} = \mathbb{E} \{ \Lambda^{-1/2}\,  E^T \, ( \mathbf{x} - \mathbf{\mu} )( \mathbf{x} - \mathbf{\mu} )^T E \, \Lambda^{-1/2}\, \}

= \Lambda^{-1/2}\,  E^T \, \mathbb{E} \{( \mathbf{x} - \mathbf{\mu} )( \mathbf{x} - \mathbf{\mu} )^T\} E \, \Lambda^{-1/2}\,

= \Lambda^{-1/2}\,  E^T \, K_{xx} E \, \Lambda^{-1/2}

By diagonalizing

K_{xx}

, we get the following:
:

\Lambda^{-1/2}\,  E^T \, E \Lambda E^T E \, \Lambda^{-1/2} = \Lambda^{-1/2}\,  \Lambda \, \Lambda^{-1/2} = I

Thus, with the above transformation, we can whiten the random vector to have zero mean and the identity covariance matrix.

RANDOM SIGNAL TRANSFORMATIONS

We can extend the same two concepts of simulating and whitening to the case of continuous time random signals or processes. For simulating, we create a filter into which we feed a white noise signal. We choose the filter so that the output signal simulates the 1st and 2nd moments of any arbitrary random process. For whitening, we feed any arbitrary random signal into a specially chosen filter so that the output of the filter is a white noise signal.

Simulating a continuous-time random signal

We can simulate any wide-sense

\mu

and Covariance function

} ight\} \mbox{ where } au = t_1 - t_2

and Power Spectral Density
:

S_x(\omega) = \int_{-\infty}^{\infty} K_x(	au) \, e^{-j \omega 	au} \, d	au

We can simulate this signal using Frequency Domain techniques.

Because

K_x(	au)

is Hermitian Symmetric and Positive Semi-definite , it follows that

S_x(\omega)

is Real and can be factored as

	CATEGORIES ABOUT WHITE NOISE

	stochastic processes

	noise

	statistics

	data compression

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