White noise

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This is a background article. See Psychological effects of exposure to 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 considered analogous to white light which contains all frequencies.

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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.

Statistical properties[]

File:Whitenoise.png

Four thousandths of a second of white noise

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. 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. 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. 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[]

Main article: Colors of noise

There are also other "colors" of noise, the most commonly used being pink, brown and blue.

Applications[]

One use for white noise is in the field of architectural acoustics. In order to mask distracting, undesirable noises in interior spaces, a constant low level of white noise is generated.

It is used by some emergency vehicle sirens due to its ability to cut through background noise and its lack of echo, which makes it easier to locate.

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 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. He or she can then adjust the overall EQ to ensure a balanced mix.

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.

White noise can be used to disorient individuals prior to interrogation and may be used as part of sensory deprivation techniques. White noise machines are sold as privacy enhancers and sleep aids and to mask tinnitus. White noise CDs, when used with headphones, can aid concentration by blocking out irritating or distracting noises in a person's environment. White Noise for Babies, such as the CD produced by Luke Rake, can aid sleep in newborns and colic suffering babies - an effective lullaby.

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 eigenvectors and $\Lambda$ is the diagonal matrix of eigenvalues.

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[]

Simulation-filter — 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 $\mu$ 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.

Because $K_{x}(\tau )$ is Hermitian symmetric and positive semi-definite, it follows that $S_{x}(\omega )$ is real and can be factored as

S_{x}(\omega )=|H(\omega )|^{2}=H(\omega )\,H^{*}(\omega )

if and only if $S_{x}(\omega )$ satisfies the Paley-Wiener criterion.

\int _{-\infty }^{\infty }{\frac {\log(S_{x}(\omega ))}{1+\omega ^{2}}}\,d\omega <\infty

If $S_{x}(\omega )$ 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 $H(\omega)$ so that its poles and zeros lie inside the left half s-plane, we can then simulate $x(t)$ with $H(\omega)$ as the transfer function of the filter.

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 $x(t)$ .

Whitening a continuous-time random signal[]

Whitening-filter — 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 $\mu$ , covariance function $K_{x}(\tau )$ , and power spectral density $S_{x}(\omega )$ as above.

We can whiten this signal using frequency domain techniques. We factor the power spectral density $S_{x}(\omega )$ as described above.

Choosing the minimum phase $H(\omega)$ 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. Additionally, we must be sure that $H(\omega)$ is strictly positive for all $\omega \in \mathbb{R}$ so that $H_{inv}(\omega )$ does not have any singularities.

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 $w(t)$ .

K_{w}(\tau )=\,\!\delta (\tau )

References & Bibliography[]

Key texts[]

Books[]

Papers[]

Additional material[]

Books[]

Rogers, L. J. (2001). Exploring the felt pathways of the self: From experience to meaning-making in children, K-5. Albany, NY: State University of New York Press.

Papers[]

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von Wright, J., & Vauras, M. (1980). Interactive effects of noise and neuroticism on recall from semantic memory: Scandinavian Journal of Psychology Vol 21(2) 1980, 97-101.
Walasek, G., & Zielinski, K. (1989). Stimulus modality effects on transformations of conditioned enhancement and conditioned suppression in rats: Acta Neurobiologiae Experimentalis 49(5) 1989, 215-238.
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Westman, A. S., & Canter, F. M. (1976). Relationship between certain personality characteristics and ability to concentrate in common noise situations: Perceptual and Motor Skills Vol 42(3, Pt 2) Jun 1976, 1259-1262.
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Dissertations[]

Arbuckle, N. J. (1978). The effect of white noise on short- and long-term recall in hyperactive boys: Dissertation Abstracts International.
Bagli, Z. R. (1978). Some latency and relaxation measures of the acoustic reflex in response to white noise and pure tones, in subjects with normal hearing and subjects with sensorineural hearing loss: Dissertation Abstracts International.
Basow, S. A. (1973). The effect of white noise on physiological arousal and attention as a function of manifest anxiety: Dissertation Abstracts International.
Canterbury, D. R. (1974). The effects of succeedingly higher level of noise upon the auditory discrimination of normal and retarded readers in third grade: Dissertation Abstracts International.
Connington, M. C. (1999). Masking overshoot: Effects of ipsilateral, bilateral and contralateral priming. Dissertation Abstracts International: Section B: The Sciences and Engineering.
Davidson, R. A. (1991). Caffeine, white noise and task novelty: Effects on electrodermal activity and task performance: Dissertation Abstracts International
Dobson, C. W. (1973). Behavior of the white-crowned sparrow (Zonotrichia leucophrys nuttalli) in a song-reinforced laboratory operant task: Dissertation Abstracts International.
Eskew, R. T. (1984). White-noise analysis of human spatial vision: Dissertation Abstracts International.
Fankhauser, C. E. (1974). An analysis of the perceptual confusions of normal hearing and hearing impaired listeners of speech masked by white noise: Dissertation Abstracts International.
Forquer, L. M. (2006). The effects of continuous white noise on the sleep patterns, mood, and cognitive performance of college students. Dissertation Abstracts International: Section B: The Sciences and Engineering.
Hammer, M. (1977). Lateral differences in the newborn infant's response to speech and noise stimuli: Dissertation Abstracts International.
Harper, D. W. (1977). A signal detection analysis of the effect of white noise intensity on visual flicker sensitivity and on response bias: Dissertation Abstracts International.
Harvey, F. W. (1981). The effects of white noise and state anxiety on cued recall: A levels of processing approach: Dissertation Abstracts International.
Horvath, T. (2008). Grooming in the rat: Novelty or stressful stimulation? Dissertation Abstracts International: Section B: The Sciences and Engineering.
Hosford, H. L. (1978). Binaural waveform coding in the inferior coliculus of the cat: Single unit responses to simple and complex stimuli: Dissertation Abstracts International.
Jazwinski, C. H. (1978). Disinhibition of verbal behavior in males and females: The effects of audience presence, audience attitude, and white noise: Dissertation Abstracts International.
Jubis, R. M. (1988). The effects of alcohol and white noise on recall of relevant and irrelevant task components: Dissertation Abstracts International.
Kim, K. L. (1980). Effects of activation and white noise-induced arousal on structural and semantic processing of verbal materials: Dissertation Abstracts International.
Legate, P. M. (1974). Effects of the timing and level of increases in arousal on performance and learning in problem-solving: Dissertation Abstracts International.
Lindsley, J. V. (1976). Changes in the concentration of power in the theta range of subicular EEG as a function of head movement and white noise stimulation: Dissertation Abstracts International.
Lorentz, R. J. (1979). The effects of accessory stimulation and training on impulsivity of high and low physiologically aroused hyperactive mentally retarded adults: Dissertation Abstracts International.
Marian, R. W. (1986). Stimulus intensity effects: A study of Stimulus Intensity Dynamism (SID) using periodic pulsed white-noise: Dissertation Abstracts International.
Millsapps, J. W. (1978). Performance of language delayed preschool children on an auditory figure-ground task: Dissertation Abstracts International.
Murphy, G. L. (1980). The effect of an aversive technique on voluntary control of heart rate: Dissertation Abstracts International.
Nearing, W. E. (1979). The effect of two levels of noise, two types of noise, and anxiety on student performance of a coding task: Dissertation Abstracts International.
Niblette, R. K. (1976). Conjugate control of stereotyped body rocking in institutionalized retarded persons using auditory stimulation: Dissertation Abstracts International.
Nozza, R. J. (1982). Detection of pure tones in quiet and in noise by infants and adults: Dissertation Abstracts International.
Onufrak, J. A. (1974). Stutterer's and nonstutterer's location of clicks superimposed on sentences of various types: Dissertation Abstracts International.
Perrillo, R. J. (1978). Effects of music, sex differences, self-efficacy expectations on reactions to failure and perceived loss of controllability: Dissertation Abstracts International.
Potash, M. (1979). The development of directional responses to sounds in the rat (Rattus norvegicus) and gerbil (Meriones unguiculatus): Dissertation Abstracts International.
Potthoff, A. D. (1979). Potentiation of Preyer Reflex sensitivity in white rats after exposure to high intensity white noise: Dissertation Abstracts International.
Rembisz, R. S. (1978). Heartbeat, performance and anxiety: Dissertation Abstracts International.
Rose, P. N. (1991). The effects of auditory noise on a peripheral visual task in a dual task paradigm: Dissertation Abstracts International.
Selters, W. A. (1974). Masked thresholds in normal and impaired ears: Dissertation Abstracts International.
Solnick, J. V. (1978). An experimental analysis of impulsivity and impulse control: Dissertation Abstracts International.
Street, S. E. (2007). Spike timing in pyramidal cells of the dorsal cochlear nucleus. Dissertation Abstracts International: Section B: The Sciences and Engineering.
Toplyn, G. A. (1988). The differential effect of noise on creative task performance: Dissertation Abstracts International.
Tubbs, R. L. (1983). The effects of intense noise and masking on pure-tone detection in rhesus monkeys (Macaca mulatta): Dissertation Abstracts International.
Venkatagiri, H. S. (1978). Experimental production and conditioning of disfluencies in the speech of normal speaking subjects using a classical conditioning paradigm: Dissertation Abstracts International.
Wang, C. (1980). Comparison of the effects of white noise and pure tone on short term verbal recall: Dissertation Abstracts International.
Webster, J. S. (1980). The effects of illumination and noise on the reaction times and tapping time in the aged: Dissertation Abstracts International.

External links[]

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White noise

Contents

Statistical properties[]

Colors of noise[]

Applications[]

Mathematical definition[]

White random vector[]

White random process (white noise)[]

Random vector transformations[]

Simulating a random vector[]

Whitening a random vector[]

Random signal transformations[]

Simulating a continuous-time random signal[]

Whitening a continuous-time random signal[]

See also[]

See also[]

References & Bibliography[]

Key texts[]

Books[]

Papers[]

Additional material[]

Books[]

Papers[]

Dissertations[]

External links[]

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