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White noise

This article is about white noise as a scientific concept, see also:

White Noise (novel) as a 1985 novel by Don DeLillo.
White Noise (movie) as a 2004 movie by Geoffrey Sax.

White noise is a signal (or process) with a flat frequency spectrum. In other words, the signal has equal power in any band, at any centre frequency, having a given bandwidth.

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

Contents

1 Statistical properties

2 Colors of noise

3 Applications

4 Whitening a random vector

5 Simulating a random vector

6 Whitening a continuous-time random signal

7 See also

8 External links

Statistical properties

A signal that is "white" in the frequency domain must have certain important statistical properties in time. For example, it must have zero autocorrelation with itself over time, except at zero timeshift. Conversely, if the autocorrelation of a signal has those properties (zero except at zero timeshift), the signal is white.

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 -1 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 (see normal distribution) is necessarily white noise. However, neither property implies the other. 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.

Colors of noise

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

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 has also been used in electronic music, where is it used either directly or as an input for a filter to create other types of noise signal.

To set up the EQ for a concert or other perfomance in a venue, white 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 and can compensate with the mixer.

Whitening a random vector

In statistics, a random vector is said to be "white" if its elements are uncorrelated and have unit variance. This corresponds to a flat power spectrum.

A vector can be whitened to remove nonzero correlations. This is useful in various procedures such as data compression.

One common method for whitening a vector $X$ with mean $μ$ is to perform the following calculation:

$X_{\mbox{white}} = \Lambda^{-1/2}\ E^T\ ( X - \mu )$

where:

X

is the vector to be whitened,

E

is the orthogonal matrix of eigenvectors of the covariance matrix $\mathbb{E}\{(X-\mu)(X-\mu)^T\}$

Λ = d i a g (λ 1,...,λ n)

is the diagonal matrix of corresponding eigenvalues

and

$\Lambda^{-1/2} = diag(\lambda_1^{-1/2},...,\lambda_n^{-1/2})$

Simulating a random vector

We can also simulate the 1st and 2nd moment properties of any vector $X$ with mean $μ$ via the following transformation of a white vector $X white$ :

$X = E\ \Lambda^{1/2}\ X_{\mbox{white}} + \mu$

Whitening a continuous-time random signal

Suppose we are given a wide-sense stationary, continuous-time random process $x(t) : t \in \mathbb{R}\,\!$ with 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 whiten this signal using frequency domain techniques.

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 .

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

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 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 $H (ω)$ is strictly positive for all $\omega \in \mathbb{R}$ so that $H i n v (ω)$ does not have any singularities.

The final form of the whitening procedure is as follows:

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

so that $x w h i t e (t)$ is mean-zero and has constant, unit power spectral density

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

Note that this power spectral density corresponds to a delta function for the covariance function of $x (t)$ .

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

External links

The colours of noise http://www.ptpart.co.uk/colors.htm
A mathematical application of noise whitening http://www.elec.qmul.ac.uk/staffinfo/markp/2003/OjaPlumbley03-ica2003.pdf in electrical engineering

Categories: Stochastic processes | Signal processing | Statistics | Signal processing | Data compression

Last updated: 02-06-2005 07:57:41

Last updated: 02-22-2005 16:12:51

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