White noise

The spectrum of noise that can be considered white


	White noise example
	A ten second snippet of audible white noise
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The characteristic “snow” on the TV screen, accompanied by white noise in the absence of a signal

White noise is stationary noise , the spectral components of which are uniformly distributed over the entire range of frequencies involved. Examples of white noise are the noise of a nearby waterfall ^[1] (the distant noise of the waterfall is pink , since the high-frequency components of the sound attenuate more strongly than the low-frequency ones in the air), or shot noise at high-resistance terminals, or the noise of a zener diode through which a very small current flows. The name received from white light containing electromagnetic waves of frequencies of the entire visible range of electromagnetic radiation. In addition to white, there are noises of many colors .

In nature and technology, “pure” white noise (that is, white noise having the same spectral power at all frequencies) does not occur (due to the fact that such a signal would have infinite power), but any noise whose spectral density falls under the category of white noise the same (or slightly different) in the considered frequency range.

Content

Statistical Properties

An example implementation of a process with white noise properties.

The term “white noise” is usually applied to a signal having an autocorrelation function mathematically described by the Dirac delta function over all dimensions of the multidimensional space in which this signal is considered. Signals with this property can be considered as white noise. This statistical property is fundamental for signals of this type.

The fact that white noise is uncorrelated in time (or in another argument) does not determine its values in the time (or any other considered argument) region . The sets received by the signal can be arbitrary up to the main statistical property (however, the constant component of such a signal must be zero). For example, the sequence of characters 1 and −1, multiplied by the sequence of delta functions following with the symbol rate, will be white noise only if the sequence of characters is uncorrelated. Signals having a continuous distribution (e.g., normal distribution ) can also be white noise.

Discrete white noise is simply a sequence of independent (i.e., not statistically related) numbers. Using the Visual C ++ package pseudo-random number generator , discrete white noise can be obtained like this:

  x [ i ] = 2 * (( rand () / (( double ) RAND_MAX )) - 0.5 )

In this case, x is an array of discrete white noise (without the zero frequency component) having a uniform distribution from −1 to 1.

It is sometimes mistakenly assumed that Gaussian noise (that is, noise with a Gaussian distribution of its values - see normal distribution ) is equivalent to white noise. However, these concepts are not equivalent. Gaussian noise involves the distribution of signal values in the form of a normal distribution, while the term “white” refers to the correlation of a signal at two different points in time (this correlation does not depend on the distribution of noise values). White noise can have any distribution, both Gaussian and Poisson , Cauchy , etc. Gaussian white noise as a model is well suited for the mathematical description of many natural processes (see Additive white Gaussian noise ).

Color noise

For convenience of description in physics, terms have been introduced that attribute various colors to noise signals depending on their statistical properties, for example, pink noise or blue noise .

Applications

White noise has many uses in physics and technology . One of them is in architectural acoustics . In order to hide unwanted noise in the interior of buildings, stationary white noise of low power is generated.

In electronic music, white noise is used both as one of the musical arrangement tools, and as an input signal for special filters that generate other types of noise signals. It is also widely used in synthesizing audio signals, usually to recreate the sound of percussion instruments such as cymbals .

Recently, many pediatricians have recommended using the sounds of white noise to calm and sleep well in infants; it is assumed that in the uterus the baby constantly heard white noise: the beat of the mother’s heart, the work of the stomach, the sound of blood in the vessels. .

White noise is used to measure the frequency characteristics of various linear dynamic systems , such as amplifiers , electronic filters , discrete control systems , etc. When applying such a system of white noise to the output, we obtain a signal that is the response of the system to the applied effect. Due to the fact that the complex frequency response of a linear system is the ratio of the Fourier transform of the output signal to the Fourier transform of the input signal, it is mathematically simple to obtain this characteristic, and for all frequencies for which the input signal can be considered white noise.

Many random number generators (both software and hardware) use white noise to generate random numbers and random sequences.

On the Linux operating system, the speaker-test console command, which generates white or pink noise , is used to test headphones / speakers.

Math Review

Random Number Vector

Random Numbers Vector $\mathbf {w}$ ${\ displaystyle \ mathbf {w}}$ is a sequence of samples of white noise when its average value $\mu _{w}$ ${\ displaystyle \ mu _ {w}}$ and autocorrelation matrix $R_{ww}$ ${\ displaystyle R_ {ww}}$ satisfy the following equalities:

\mu _{w}=\mathbb {E} \{\mathbf {w} \}=0

{\ displaystyle \ mu _ {w} = \ mathbb {E} \ {\ mathbf {w} \} = 0}

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

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

That is, this is a vector of random numbers with a zero mean value, the autocorrelation matrix of which is a diagonal matrix with dispersions along the main diagonal.

White random process (white noise)

Time-continuous random process $w(t)$ ${\ displaystyle w (t)}$ where $t\in \mathbb {R}$ ${\ displaystyle t \ in \ mathbb {R}}$ is white noise if and only if its mathematical expectation and autocorrelation function satisfy the following equalities, respectively:

\mu _{w}(t)=\mathbb {E} \{w(t)\}=0

{\ displaystyle \ 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})

{\ displaystyle R_ {ww} (t_ {1}, t_ {2}) = \ mathbb {E} \ {w (t_ {1}) w (t_ {2}) \} = \ sigma ^ {2} \ delta (t_ {1} -t_ {2})}

.

If the value $\sigma ^{2}$ ${\ displaystyle \ sigma ^ {2}}$ does not depend on time, then the random process is stationary white noise , if it depends on time - non - stationary white noise ^[2] .

In other notations closer to the radiophysicists of the national school:

\langle w(t)\rangle =0{\frac {}{}}

{\ displaystyle \ langle w (t) \ rangle = 0 {\ frac {} {}}}

B_{ww}(t_{1},t_{2})\equiv \langle \,[w(t_{1})-\langle w(t_{1})\rangle ]\,[w(t_{2})-\langle w(t_{2})\rangle ]\,\rangle =\langle \,w(t_{1})w(t_{2})\,\rangle =\sigma _{w}^{2}\delta (t_{1}-t_{2})

{\ displaystyle B_ {ww} (t_ {1}, t_ {2}) \ equiv \ langle \, [w (t_ {1}) - \ langle w (t_ {1}) \ rangle] \, [w ( t_ {2}) - \ langle w (t_ {2}) \ rangle] \, \ rangle = \ langle \, w (t_ {1}) w (t_ {2}) \, \ rangle = \ sigma _ { w} ^ {2} \ delta (t_ {1} -t_ {2})}

.

That is, it is a random process with zero mathematical expectation, which has an autocorrelation function , which is the Dirac delta function . Such an autocorrelation function assumes the following power spectral density :

S_{ww}(\omega )=\sigma _{w}^{2}

{\ displaystyle S_ {ww} (\ omega) = \ sigma _ {w} ^ {2}}

since the Fourier transform of the delta function is equal to unity at all frequencies. Due to the fact that the power spectral density is the same at all frequencies, white noise got its name (by analogy with the white light frequency spectrum).

Notes

↑ TSB
↑ Ventzel E.S. , Ovcharov L.A. Theory of random processes and its engineering applications. - M., Science, 1991. - c. 274

Literature

White noise // Big Russian Encyclopedia : [in 35 vols.] / Ch. ed. Yu.S. Osipov . - M .: Great Russian Encyclopedia, 2004—2017.
White noise in probability theory / Yu. V. Prokhorov // Big Russian Encyclopedia : [in 35 vols.] / Ch. ed. Yu.S. Osipov . - M .: Great Russian Encyclopedia, 2004—2017.

[1] TSB

[2] Ventzel E.S. , Ovcharov L.A. Theory of random processes and its engineering applications. - M., Science, 1991. - c. 274