White noise

Colors of noise
White Pink Red (Brownian) Purple Grey
v t e

In

.

In

The samples of a white noise signal may be

.

An infinite-bandwidth white noise signal is a purely theoretical construction. The bandwidth of white noise is limited in practice by the mechanism of noise generation, by the transmission medium and by finite observation capabilities. Thus, random signals are considered "white noise" if they are observed to have a flat spectrum over the range of frequencies that are relevant to the context. For an audio signal, the relevant range is the band of audible sound frequencies (between 20 and 20,000 Hz). Such a signal is heard by the human ear as a hissing sound, resembling the /h/ sound in a sustained aspiration. On the other hand, the "sh" sound /ʃ/ in "ash" is a colored noise because it has a formant structure. In music and acoustics, the term "white noise" may be used for any signal that has a similar hissing sound.

The term white noise is sometimes used in the context of phylogenetically based statistical methods to refer to a lack of phylogenetic pattern in comparative data.^[5] It is sometimes used analogously in nontechnical contexts to mean "random talk without meaningful contents".^[6][7]

Statistical properties

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Any distribution of values is possible (although it must have zero

, can of course be white.

It is often incorrectly assumed that Gaussian noise (i.e., noise with a Gaussian amplitude distribution – see normal distribution) necessarily refers to white noise, yet neither property implies the other. Gaussianity refers to the probability distribution with respect to the value, in this context the probability of the signal falling within any particular range of amplitudes, while the term 'white' refers to the way the signal power is distributed (i.e., independently) over time or among frequencies.

One form of white noise is the generalized mean-square derivative of the Wiener process or Brownian motion.

A generalization to random elements on infinite dimensional spaces, such as random fields, is the white noise measure.

Practical applications

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Music

White noise is commonly used in the production of

A simple example of white noise is a nonexistent radio station (static).

Electronics engineering

White noise is also used to obtain the impulse response of an electrical circuit, in particular of amplifiers and other audio equipment. It is not used for testing loudspeakers as its spectrum contains too great an amount of high-frequency content. Pink noise, which differs from white noise in that it has equal energy in each octave, is used for testing transducers such as loudspeakers and microphones.

Computing

White noise is used as the basis of some random number generators. For example, Random.org uses a system of atmospheric antennas to generate random digit patterns from sources that can be well-modeled by white noise.^[9]

Tinnitus treatment

White noise is a common synthetic noise source used for sound masking by a tinnitus masker.^[10] White noise machines and other white noise sources are sold as privacy enhancers and sleep aids (see music and sleep) and to mask tinnitus.^[11] The Marpac Sleep-Mate was the first domestic use white noise machine built in 1962 by traveling salesman Jim Buckwalter.^[12] Alternatively, the use of an FM radio tuned to unused frequencies ("static") is a simpler and more cost-effective source of white noise.^[13] However, white noise generated from a common commercial radio receiver tuned to an unused frequency is extremely vulnerable to being contaminated with spurious signals, such as adjacent radio stations, harmonics from non-adjacent radio stations, electrical equipment in the vicinity of the receiving antenna causing interference, or even atmospheric events such as solar flares and especially lightning.

Work environment

The effects of white noise upon cognitive function are mixed. Recently, a small study found that white noise background stimulation improves cognitive functioning among secondary students with attention deficit hyperactivity disorder (ADHD), while decreasing performance of non-ADHD students.^[14][15] Other work indicates it is effective in improving the mood and performance of workers by masking background office noise,^[16] but decreases cognitive performance in complex card sorting tasks.^[17]

Similarly, an experiment was carried out on sixty-six healthy participants to observe the benefits of using white noise in a learning environment. The experiment involved the participants identifying different images whilst having different sounds in the background. Overall the experiment showed that white noise does in fact have benefits in relation to learning. The experiments showed that white noise improved the participants' learning abilities and their recognition memory slightly.^[18]

Mathematical definitions

White noise vector

A

A necessary (but,

correlation

matrix must be the n by n identity matrix.

If, in addition to being independent, every variable in w also has a normal distribution with zero mean and the same variance ${\displaystyle \sigma ^{2}}$, w is said to be a Gaussian white noise vector. In that case, the joint distribution of w is a multivariate normal distribution; the independence between the variables then implies that the distribution has spherical symmetry in n-dimensional space. Therefore, any orthogonal transformation of the vector will result in a Gaussian white random vector. In particular, under most types of discrete Fourier transform, such as FFT and Hartley, the transform W of w will be a Gaussian white noise vector, too; that is, the n Fourier coefficients of w will be independent Gaussian variables with zero mean and the same variance ${\displaystyle \sigma ^{2}}$.

The

of each coefficient of its Fourier transform W, that is, P_i = E(|W_i|²). Under that definition, a Gaussian white noise vector will have a perfectly flat power spectrum, with P_i = σ² for all i.

If w is a white random vector, but not a Gaussian one, its Fourier coefficients W_i will not be completely independent of each other; although for large n and common probability distributions the dependencies are very subtle, and their pairwise correlations can be assumed to be zero.

Often the weaker condition "statistically uncorrelated" is used in the definition of white noise, instead of "statistically independent". However, some of the commonly expected properties of white noise (such as flat power spectrum) may not hold for this weaker version. Under this assumption, the stricter version can be referred to explicitly as independent white noise vector.^[20]: p.60 Other authors use strongly white and weakly white instead.^[21]

An example of a random vector that is "Gaussian white noise" in the weak but not in the strong sense is ${\displaystyle x=[x_{1},x_{2}]}$ where ${\displaystyle x_{1}}$ is a normal random variable with zero mean, and ${\displaystyle x_{2}}$ is equal to ${\displaystyle +x_{1}}$ or to ${\displaystyle -x_{1}}$, with equal probability. These two variables are uncorrelated and individually normally distributed, but they are not jointly normally distributed and are not independent. If ${\displaystyle x}$ is rotated by 45 degrees, its two components will still be uncorrelated, but their distribution will no longer be normal.

In some situations, one may relax the definition by allowing each component of a white random vector ${\displaystyle w}$ to have non-zero expected value ${\displaystyle \mu }$. In

especially, where samples are typically restricted to positive values, one often takes ${\displaystyle \mu }$ to be one half of the maximum sample value. In that case, the Fourier coefficient ${\displaystyle W_{0}}$ corresponding to the zero-frequency component (essentially, the average of the ${\displaystyle w_{i}}$) will also have a non-zero expected value ${\displaystyle \mu {\sqrt {n}}}$; and the power spectrum ${\displaystyle P}$ will be flat only over the non-zero frequencies.

Discrete-time white noise

A discrete-time stochastic process ${\displaystyle W(n)}$ is a generalization of a random vector with a finite number of components to infinitely many components. A discrete-time stochastic process ${\displaystyle W(n)}$ is called white noise if its mean is equal to zero for all ${\displaystyle n}$ , i.e. ${\displaystyle \operatorname {E} [W(n)]=0}$ and if the autocorrelation function ${\displaystyle R_{W}(n)=\operatorname {E} [W(k+n)W(k)]}$ has a nonzero value only for ${\displaystyle n=0}$, i.e. ${\displaystyle R_{W}(n)=\sigma ^{2}\delta (n)}$.^{[citation needed][clarification needed]}

Continuous-time white noise

In order to define the notion of "white noise" in the theory of

signals, one must replace the concept of a "random vector" by a continuous-time random signal; that is, a random process that generates a function ${\displaystyle w}$ of a real-valued parameter ${\displaystyle t}$.

Such a process is said to be white noise in the strongest sense if the value ${\displaystyle w(t)}$ for any time ${\displaystyle t}$ is a random variable that is statistically independent of its entire history before ${\displaystyle t}$. A weaker definition requires independence only between the values ${\displaystyle w(t_{1})}$ and ${\displaystyle w(t_{2})}$ at every pair of distinct times ${\displaystyle t_{1}}$ and ${\displaystyle t_{2}}$. An even weaker definition requires only that such pairs ${\displaystyle w(t_{1})}$ and ${\displaystyle w(t_{2})}$ be uncorrelated.^[22] As in the discrete case, some authors adopt the weaker definition for "white noise", and use the qualifier independent to refer to either of the stronger definitions. Others use weakly white and strongly white to distinguish between them.

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 ${\displaystyle w}$ is no longer a finite-dimensional space ${\displaystyle \mathbb {R} ^{n}}$, but an infinite-dimensional function space. Moreover, by any definition a white noise signal ${\displaystyle w}$ would have to be essentially discontinuous at every point; therefore even the simplest operations on ${\displaystyle w}$, like integration over a finite interval, require advanced mathematical machinery.

Some authors^{[citation needed][clarification needed]} require each value ${\displaystyle w(t)}$ to be a real-valued random variable with expectation ${\displaystyle \mu }$ and some finite variance ${\displaystyle \sigma ^{2}}$. Then the covariance ${\displaystyle \mathrm {E} (w(t_{1})\cdot w(t_{2}))}$ between the values at two times ${\displaystyle t_{1}}$ and ${\displaystyle t_{2}}$ is well-defined: it is zero if the times are distinct, and ${\displaystyle \sigma ^{2}}$ if they are equal. However, by this definition, the integral

${\displaystyle W_{[a,a+r]}=\int _{a}^{a+r}w(t)\,dt}$

over any interval with positive width ${\displaystyle r}$ would be simply the width times the expectation: ${\displaystyle r\mu }$.^{[clarification needed]} This property renders the concept inadequate as a model of "white noise" signals either in a physical or mathematical sense.^{[clarification needed]}

Therefore, most authors define the signal ${\displaystyle w}$ indirectly by specifying random values for the integrals of ${\displaystyle w(t)}$ and ${\displaystyle |w(t)|^{2}}$ over each interval ${\displaystyle [a,a+r]}$. In this approach, however, the value of ${\displaystyle w(t)}$ at an isolated time cannot be defined as a real-valued random variable^{[citation needed]}. Also the covariance ${\displaystyle \mathrm {E} (w(t_{1})\cdot w(t_{2}))}$ becomes infinite when ${\displaystyle t_{1}=t_{2}}$; and the autocorrelation function ${\displaystyle \mathrm {R} (t_{1},t_{2})}$ must be defined as ${\displaystyle N\delta (t_{1}-t_{2})}$, where ${\displaystyle N}$ is some real constant and ${\displaystyle \delta }$ is Dirac's "function".^{[clarification needed]}

In this approach, one usually specifies that the integral ${\displaystyle W_{I}}$ of ${\displaystyle w(t)}$ over an interval ${\displaystyle I=[a,b]}$ is a real random variable with normal distribution, zero mean, and variance ${\displaystyle (b-a)\sigma ^{2}}$; and also that the covariance ${\displaystyle \mathrm {E} (W_{I}\cdot W_{J})}$ of the integrals ${\displaystyle W_{I}}$, ${\displaystyle W_{J}}$ is ${\displaystyle r\sigma ^{2}}$, where ${\displaystyle r}$ is the width of the intersection ${\displaystyle I\cap J}$ of the two intervals ${\displaystyle I,J}$. This model is called a Gaussian white noise signal (or process).

In the mathematical field known as white noise analysis, a Gaussian white noise ${\displaystyle w}$ is defined as a stochastic tempered distribution, i.e. a random variable with values in the space ${\displaystyle {\mathcal {S}}'(\mathbb {R} )}$ of tempered distributions. Analogous to the case for finite-dimensional random vectors, a probability law on the infinite-dimensional space ${\displaystyle {\mathcal {S}}'(\mathbb {R} )}$ can be defined via its characteristic function (existence and uniqueness are guaranteed by an extension of the Bochner–Minlos theorem, which goes under the name Bochner–Minlos–Sazanov theorem); analogously to the case of the multivariate normal distribution ${\displaystyle X\sim {\mathcal {N}}_{n}(\mu ,\Sigma )}$, which has characteristic function

${\displaystyle \forall k\in \mathbb {R} ^{n}:\quad \mathrm {E} (\mathrm {e} ^{\mathrm {i} \langle k,X\rangle })=\mathrm {e} ^{\mathrm {i} \langle k,\mu \rangle -{\frac {1}{2}}\langle k,\Sigma k\rangle },}$

the white noise ${\displaystyle w:\Omega \to {\mathcal {S}}'(\mathbb {R} )}$ must satisfy

${\displaystyle \forall \varphi \in {\mathcal {S}}(\mathbb {R} ):\quad \mathrm {E} (\mathrm {e} ^{\mathrm {i} \langle w,\varphi \rangle })=\mathrm {e} ^{-{\frac {1}{2}}\|\varphi \|_{2}^{2}},}$

where ${\displaystyle \langle w,\varphi \rangle }$ is the natural pairing of the tempered distribution ${\displaystyle w(\omega )}$ with the Schwartz function ${\displaystyle \varphi }$, taken scenariowise for ${\displaystyle \omega \in \Omega }$, and ${\displaystyle \|\varphi \|_{2}^{2}=\int _{\mathbb {R} }\vert \varphi (x)\vert ^{2}\,\mathrm {d} x}$.

Mathematical applications

Time series analysis and regression

In

 – that is, if it has different variances for different data points.

Alternatively, in the subset of regression analysis known as

process, in which the current value of the dependent variable depends on current and past values of a sequential white noise process.

Random vector transformations

These two ideas are crucial in applications such as

.

In particular, by a suitable linear transformation (a coloring transformation), a white random vector can be used to produce a "non-white" random vector (that is, a list of random variables) whose elements have a prescribed covariance matrix. Conversely, a random vector with known covariance matrix can be transformed into a white random vector by a suitable whitening transformation.

Generation

White noise may be generated digitally with a digital signal processor, microprocessor, or microcontroller. Generating white noise typically entails feeding an appropriate stream of random numbers to a digital-to-analog converter. The quality of the white noise will depend on the quality of the algorithm used.^[23]

Informal use

The term is sometimes used as a colloquialism to describe a backdrop of ambient sound, creating an indistinct or seamless commotion. Following are some examples:

• Chatter from multiple conversations within the acoustics of a confined space.
• The pleonastic jargon used by politicians to mask a point that they don't want noticed.^[24]
• Music that is disagreeable, harsh, dissonant or discordant with no melody.

The term can also be used metaphorically, as in the novel White Noise (1985) by Don DeLillo which explores the symptoms of modern culture that came together so as to make it difficult for an individual to actualize their ideas and personality.

See also

Colors of noise
White Pink Red (Brownian) Purple Grey
v t e

References

External links

Wikimedia Commons has media related to White noise.