White noise

For other uses, see White noise (disambiguation).

Colors of noise
White
Pink
Brown/Red
Grey

Calculated spectrum of a generated approximation of white noise

White noise is a random signal (or process) with a flat power spectral density. Even though Noise is a random Signal, it can have characteristic statistical properties Pink noise or 1/f noise is a signal or process with a Frequency spectrum such that the power spectral density is Proportional In Science, Brownian noise ( also known as Brown noise or red noise, is the kind of Signal noise produced by Brownian motion Grey noise is random noise subjected to a Psychoacoustic Equal loudness curve (such as an inverted A-weighting curve over In the fields of communications, Signal processing, and in Electrical engineering more generally a signal is any time-varying or spatial-varying quantity In Statistical signal processing and Physics, the spectral density, power spectral density ( PSD) or energy 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 considered analogous to white light which contains all frequencies. White is a Color, the perception which is evoked by Light that stimulates all three types of color sensitive Cone cells in the Human eye

An infinite-bandwidth, white noise signal is purely a theoretical construction. 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.

1 Statistical properties
2 Applications
3 Mathematical definition
- 3.1 White random vector
- 3.2 White random process (white noise)
4 Random vector transformations
- 4.1 Simulating a random vector
- 4.2 Whitening a random vector
5 Random signal transformations
- 5.1 Simulating a continuous-time random signal
- 5.2 Whitening a continuous-time random signal
6 See also
7 External links

Statistical properties

An example realization of a Gaussian white noise process.

The term white noise is also commonly applied to a noise signal in the spatial domain which has an autocorrelation which can be represented by a delta function over the relevant space dimensions. Autocorrelation is a mathematical tool for finding repeating patterns such as the presence of a periodic signal which has been buried under noise or identifying the Missing fundamental The signal is then "white" in the spatial frequency domain (this is equally true for signals in the angular frequency domain, e. In Mathematics, Physics, and Engineering, spatial frequency is a characteristic of any structure that is periodic across position in space 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, on Linux white noise can be generated with the command cat /dev/urandom > /dev/dsp, feeding the kernel random number generator (uniformly distributed integers between 0 and 255) into the digital signal processor. Linux (commonly pronounced ˈlɪnəks The cat command is a standard Unix program used to concatenate and display files In Unix-like Operating systems /dev/random is a Special file that serves as a true Random number generator or as a Pseudorandom number generator Even 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. The normal distribution, also called the Gaussian distribution, is an important family of Continuous probability distributions applicable in many fields

It is often incorrectly assumed that Gaussian noise (i. Gaussian noise is noise that has a Probability density function (abbreviated pdf of the Normal distribution (also known as Gaussian distribution e. , noise with a Gaussian amplitude distribution — see normal distribution) is necessarily white noise. The normal distribution, also called the Gaussian distribution, is an important family of Continuous probability distributions applicable in many fields However, neither property implies the other. Gaussianity refers to the probability that the signal has a certain value at a certain instant, while the term 'white' refers to the way the signal power (taken over time) is distributed among frequencies.

Pink noise (left) and white noise (right) on a FFT spectrogram with linear frequency axis (vertical)

We can therefore find Gaussian white noise, but also Poisson, Cauchy, etc. Pink noise or 1/f noise is a signal or process with a Frequency spectrum such that the power spectral density is Proportional The spectrogram is the result of calculating the Frequency spectrum of Windowed frames of a compound signal. white noises. 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 additive white Gaussian noise has a standard abbreviation: AWGN. Explanation In communications, the additive white Gaussian noise ( AWGN) channel model is one in which the only impairment is the linear addition of Explanation In communications, the additive white Gaussian noise ( AWGN) channel model is one in which the only impairment is the linear addition of Gaussian white noise has the useful statistical property that its values are independent (see Statistical independence). In Probability theory, to say that two events are independent intuitively means that the occurrence of one event makes it neither more nor less probable that the other

White noise is the generalized mean-square derivative of the Wiener process or Brownian motion. In Mathematics, the Wiener process is a continuous-time Stochastic process named in honor of Norbert Wiener. This article is about the physical phenomenon for the stochastic process see Wiener process.

Applications

One use for white noise is in the field of architectural acoustics. Architectural acoustics is the science of controlling sound within buildings In order to dissemble distracting, undesirable noises in interior spaces, a low level of constant white noise is generated.

It is used by some emergency vehicle sirens due to its ability to cut through background noise, which makes it easier to locate. A siren is a loud noise maker The original version would yield sounds under water suggesting a link with the Sirens of Greek mythology

White noise is commonly used in the production of electronic music, usually either directly or as an input for a filter to create other types of noise signal. Electronic music is music that employs Electronic musical instruments and Electronic Music technology in its production It is used extensively in audio synthesis, typically to recreate percussive instruments such as cymbals which have high noise content in their frequency domain. Cymbals are a modern percussion instrument Cymbals consist of thin normally round plates of various Cymbal alloys; see Cymbal making for a discussion of their

It is also used to generate impulse responses. The impulse response of a system is its output when presented with a very brief input signal an impulse To set up the EQ for a concert or other performance in a venue, a short burst of white or 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. Equalization (or equalisation, EQ) is the process of changing the frequency envelope of a sound in Audio processing. The engineer can then adjust the overall EQ to ensure a balanced mix.

Music sample:

White noise

10 second sample of white sound.

Problems listening to the file? See media help.

White noise can be used for frequency response testing of amplifiers and electronic filters. It is sometimes used with a flat response microphone and an automatic equalizer. The idea is that the system will generate white noise and the microphone will pick up the white noise produced by the speakers. It will then automatically equalize each frequency band to get a flat response. That system is used in professional level equipment, some high-end home stereo and some high-end car radios.

White noise is used as the basis of some random number generators. In Computing, a hardware random number generator is an apparatus that generates Random numbers from a physical process

White noise can be used to disorient individuals prior to interrogation and may be used as part of sensory deprivation techniques. Interrogation or questioning is Interviewing as commonly employed by officers of the Police and Military. Sensory deprivation is the deliberate reduction or removal of stimuli from one or more of the senses White noise machines are sold as privacy enhancers and sleep aids and to mask tinnitus. white noise machine is a device that produces a sound that is random in character somewhat like a waterfall or air escaping from a balloon Tinnitus (tɪˈnaɪtəs or /ˈtɪnɪtəs/ from the Latin word for " Ringing " is the perception of sound within the human ear in the absence of corresponding White noise CDs, when used with headphones, can aid concentration by blocking out irritating or distracting noises in a person's environment.

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:

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

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

That is, it is a zero mean random vector, and its autocorrelation matrix is a multiple of the identity matrix. In Statistics, mean has two related meanings the Arithmetic mean (and is distinguished from the Geometric mean or Harmonic mean Autocorrelation is a mathematical tool for finding repeating patterns such as the presence of a periodic signal which has been buried under noise or identifying the Missing fundamental In Linear algebra, the identity matrix or unit matrix of size n is the n -by- n Square matrix with ones on the Main 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:

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

$R_{ww}(t_1, t_2) = \mathbb{E}\{ w(t_1) w(t_2)\} = (N_{0}/2)\delta(t_1 - t_2)$ .

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 Dirac delta or Dirac's delta is a mathematical construct introduced by the British theoretical physicist Paul Dirac.

The above autocorrelation function implies the following power spectral density.

$S_{xx}(\omega) = N_{0}/2 ,\!$

since the Fourier transform of the delta function and likewise is equal to 1. This article specifically discusses Fourier transformation of functions on the Real line; for other kinds of Fourier transformation see Fourier analysis and 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. In Statistical signal processing and Physics, the spectral density, power spectral density ( PSD) or energy spectral density ( Familiar concepts associated with a Frequency are colors musical notes radio/TV channels and even the regular rotation of the earth White is a Color, the perception which is evoked by Light that stimulates all three types of color sensitive Cone cells in the Human eye

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. In Statistics and Probability theory, the covariance matrix is a matrix of Covariances between elements of a vector In Statistics and Probability theory, the covariance matrix is a matrix of Covariances between elements of a vector 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 $K x x$ . In Statistics and Probability theory, the covariance matrix is a matrix of Covariances between elements of a vector 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. In Mathematics, specifically in Functional analysis, each Linear operator on a Hilbert space has a corresponding adjoint operator. In mathematics positive semidefinite may refer to Positive-semidefinite matrix Positive-semidefinite function In Mathematics, particularly Linear algebra and Functional analysis, the spectral theorem is any of a number of results about Linear operators Linear algebra is the branch of Mathematics concerned with

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

where $E$ is the orthogonal matrix of eigenvectors and $Λ$ is the diagonal matrix of eigenvalues. In Matrix theory, a real orthogonal matrix is a square matrix Q whose Transpose is its inverse: Q^T In Mathematics, given a Linear transformation, an of that linear transformation is a nonzero vector which when that transformation is applied to it changes In Linear algebra, a diagonal matrix is a Square matrix in which the entries outside the Main diagonal (↘ are all zero In Mathematics, given a Linear transformation, an of that linear transformation is a nonzero vector which when that transformation is applied to it changes

We can simulate the 1st and 2nd moment properties of this random vector $\mathbf{x}$ with mean $\mathbf{\mu}$ and covariance matrix $K x x$ 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 $K x x$ 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 $K x x$ , 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. A multivariate random variable or random vector is a vector X = ( X 1. In Statistics, mean has two related meanings the Arithmetic mean (and is distinguished from the Geometric mean or Harmonic mean In Statistics, mean has two related meanings the Arithmetic mean (and is distinguished from the Geometric mean or Harmonic mean In Statistics and Probability theory, the covariance matrix is a matrix of Covariances between elements of a vector

Random signal transformations

We cannot 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

White noise fed into a linear, time-invariant filter to simulate the 1st and 2nd moments of an arbitrary random process.

We can simulate any wide-sense stationary, continuous-time random process $x(t) : t \in \mathbb{R}\,\!$ with constant mean $μ$ and covariance function

$K_x(\tau) = \mathbb{E} \left\{ (x(t_1) - \mu) (x(t_2) - \mu)^{*} \right\} \mbox{ where } \tau = t_1 - t_2$

and power spectral density

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

We can simulate this signal using frequency domain techniques. In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output A stochastic process, or sometimes random process, is the counterpart to a deterministic process (or Deterministic system) in Probability theory. In Statistics, mean has two related meanings the Arithmetic mean (and is distinguished from the Geometric mean or Harmonic mean In Probability theory and Statistics, covariance is a measure of how much two variables change together (the Variance is a special case of the covariance In Statistical signal processing and Physics, the spectral density, power spectral density ( PSD) or energy spectral density ( Frequency domain is a term used to describe the analysis of Mathematical functions or signals with respect to frequency

Because $K x (τ)$ is Hermitian symmetric and positive semi-definite, it follows that $S x (ω)$ is real and can be factored as

$S_x(\omega) = | H(\omega) |^2 = H(\omega) \, H^{*} (\omega)$

if and only if $S x (ω)$ satisfies the Paley-Wiener criterion. A number of Mathematical entities are named Hermitian, after the Mathematician Charles Hermite: Hermitian adjoint In Mathematics, a definite bilinear form is a Bilinear form B such that B ( x, x) has a fixed In Mathematics, the real numbers may be described informally in several different ways

$\int_{-\infty}^{\infty} \frac{\log (S_x(\omega))}{1 + \omega^2} \, d \omega < \infty$

If $S x (ω)$ is a rational function, we can then factor it into pole-zero form as

$S_x(\omega) = \frac{\Pi_{k=1}^{N} (c_k - j \omega)(c^{*}_k + j \omega)}{\Pi_{k=1}^{D} (d_k - j \omega)(d^{*}_k + j \omega)}$

Choosing a minimum phase $H (ω)$ so that its poles and zeros lie inside the left half s-plane, we can then simulate $x (t)$ with $H (ω)$ as the transfer function of the filter. In Mathematics, a rational function is any function which can be written as the Ratio of two Polynomial functions Definitions In In Complex analysis, a pole of a Meromorphic function is a certain type of singularity that behaves like the singularity at z = 0 In Complex analysis, a zero of a Holomorphic function f is a Complex number a such that f ( a) = 0 In Control theory and Signal processing, a linear time-invariant system is said to be minimum-phase if the system and its inverse are causal The S plane is a mathematical domain where instead of viewing processes in the time domain modelled with time-based functions they are viewed as equations in the frequency domain

We can simulate $x (t)$ by constructing the following linear, time-invariant filter

$\hat{x}(t) = \mathcal{F}^{-1} \left\{ H(\omega) \right\} * w(t) + \mu$

where $w (t)$ is a continuous-time, white-noise signal with the following 1st and 2nd moment properties:

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

$\mathbb{E}\{w(t_1)w^{*}(t_2)\} = K_w(t_1, t_2) = \delta(t_1 - t_2)$

Thus, the resultant signal $\hat{x}(t)$ has the same 2nd moment properties as the desired signal $x (t)$ . The word linear comes from the Latin word linearis, which means created by lines. A time-invariant system is one whose output does not depend explicitly on time Electronic filters are Electronic circuits which perform Signal processing functions specifically intended to remove unwanted signal components and/or enhance wanted In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output

Whitening a continuous-time random signal

An arbitrary random process x(t) fed into a linear, time-invariant filter that whitens x(t) to create white noise at the output.

Suppose we have a wide-sense stationary, continuous-time random process $x(t) : t \in \mathbb{R}\,\!$ defined with the same mean $μ$ , covariance function $K x (τ)$ , and power spectral density $S x (ω)$ as above. In the mathematical sciences, a stationary process (or strict(ly stationary process or strong(ly stationary process) is a Stochastic process In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output A stochastic process, or sometimes random process, is the counterpart to a deterministic process (or Deterministic system) in Probability theory. In Statistics, mean has two related meanings the Arithmetic mean (and is distinguished from the Geometric mean or Harmonic mean In Probability theory and Statistics, covariance is a measure of how much two variables change together (the Variance is a special case of the covariance In Statistical signal processing and Physics, the spectral density, power spectral density ( PSD) or energy spectral density (

We can whiten this signal using frequency domain techniques. Frequency domain is a term used to describe the analysis of Mathematical functions or signals with respect to frequency We factor the power spectral density $S x (ω)$ as described above.

Choosing the minimum phase $H (ω)$ so that its poles and zeros lie inside the left half s-plane, we can then whiten $x (t)$ with the following inverse filter

$H_{inv}(\omega) = \frac{1}{H(\omega)}$

We choose the minimum phase filter so that the resulting inverse filter is stable. In Control theory and Signal processing, a linear time-invariant system is said to be minimum-phase if the system and its inverse are causal The S plane is a mathematical domain where instead of viewing processes in the time domain modelled with time-based functions they are viewed as equations in the frequency domain In Control theory and Signal processing, a linear time-invariant system is said to be minimum-phase if the system and its inverse are causal Bibo redirects here For the Egyptian football player nicknamed Bibo see Mahmoud El-Khateeb. Additionally, we must be sure that $H (ω)$ is strictly positive for all $\omega \in \mathbb{R}$ so that $H i n v (ω)$ does not have any singularities. In Mathematics, a singularity is in general a point at which a given mathematical object is not defined or a point of an exceptional set where it fails to be

The final form of the whitening procedure is as follows:

$w (t) = \mathcal{F}^{-1} \left\{ H_{inv}(\omega) \right\} * (x(t) - \mu)$

so that $w (t)$ is a white noise random process with zero mean and constant, unit power spectral density

$S_{w}(\omega) = \mathcal{F} \left\{ \mathbb{E} \{ w(t_1) w(t_2) \} \right\} = H_{inv}(\omega) S_x(\omega) H^{*}_{inv}(\omega) = \frac{S_x(\omega)}{S_x(\omega)} = 1$

Note that this power spectral density corresponds to a delta function for the covariance function of $w (t)$ . A stochastic process, or sometimes random process, is the counterpart to a deterministic process (or Deterministic system) in Probability theory. In Statistics, mean has two related meanings the Arithmetic mean (and is distinguished from the Geometric mean or Harmonic mean In Statistical signal processing and Physics, the spectral density, power spectral density ( PSD) or energy spectral density ( In Statistical signal processing and Physics, the spectral density, power spectral density ( PSD) or energy spectral density ( In Probability theory and Statistics, covariance is a measure of how much two variables change together (the Variance is a special case of the covariance

$K_w(\tau) = \,\!\delta (\tau)$

External links

Even though Noise is a random Signal, it can have characteristic statistical properties Electronics refers to the flow of charge (moving Electrons through Nonmetal conductors (mainly Semiconductors, whereas electrical Electronic noise is an unwanted signal characteristic of all electronic circuits. Independent component analysis ( ICA) is a computational method for separating a Multivariate signal into additive subcomponents supposing the mutual Statistical In Science, and especially in Physics and Telecommunication, noise is fluctuations in and the addition of external factors to the stream of target Statistics is a mathematical science pertaining to the collection analysis interpretation or explanation and presentation of Data. white noise machine is a device that produces a sound that is random in character somewhat like a waterfall or air escaping from a balloon Architectural acoustics is the science of controlling sound within buildings Sound masking is the addition of natural or artificial sound (commonly though inaccurately referred to as " White noise " or " Pink noise " Pink noise or 1/f noise is a signal or process with a Frequency spectrum such that the power spectral density is Proportional In Science, Brownian noise ( also known as Brown noise or red noise, is the kind of Signal noise produced by Brownian motion

Dictionary

-noun

(physics) A random signal (or process) with a flat power spectral density; a signal with a power spectral density that has equal power in any band, at any centre frequency, having a given bandwidth.
(nontechnically) Any nondescript noise used for background or to mask or drown out other noise.

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