Editing White noise (section)

===White noise vector===
A [[random vector]] (that is, a random variable with values in ''R<sup>n</sup>'') is said to be a white noise vector or white random vector if its components each have a [[probability distribution]] with zero mean and finite [[variance]],{{what|reason=Why aren't the variances required to be identical, like in the Gaussian case (two paragraphs below) and the discrete-time case (next section)?|date=October 2023}}  and are [[statistically independent]]: that is, their [[joint probability distribution]] must be the product of the distributions of the individual components.<ref name="fessler">
  Jeffrey A. Fessler (1998), ''[https://web.archive.org/web/20131218214647/http://andywilliamson.org/_/wp-content/uploads/2010/04/White-Noise.pdf On Transformations of Random Vectors.]'' Technical report 314, Dept. of Electrical Engineering and Computer Science, Univ. of Michigan. ([[PDF]])</ref>

A necessary (but, [[normally distributed and uncorrelated does not imply independent|in general, not sufficient]]) condition for statistical independence of two variables is that they be [[correlation|statistically uncorrelated]]; that is, their [[covariance]] is zero. Therefore, the  [[covariance matrix]] ''R'' of the components of a white noise vector ''w'' with ''n'' elements must be an ''n'' by ''n'' [[diagonal matrix]], where each diagonal element ''R<sub>ii</sub>'' is the [[variance]] of component ''w<sub>i</sub>''; and the [[Correlation and dependence#Correlation matrices|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 <math>\sigma^2</math>, ''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 [[elliptical distribution|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 [[discrete Hartley transform|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 <math>\sigma^2</math>.

The [[power spectrum]] ''P'' of a random vector ''w'' can be defined as the expected value of the [[squared modulus]] of each coefficient of its Fourier transform ''W'', that is, ''P<sub>i</sub>'' = E(|''W<sub>i</sub>''|<sup>2</sup>). Under that definition, a Gaussian white noise vector will have a perfectly flat power spectrum, with ''P<sub>i</sub>''&nbsp;=&nbsp;''σ''<sup>2</sup> for all&nbsp;''i''.

If ''w'' is a white random vector, but not a Gaussian one, its Fourier coefficients ''W<sub>i</sub>'' 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.<ref name="ezivot">Eric Zivot and Jiahui Wang (2006), [http://faculty.washington.edu/ezivot/econ584/notes/timeSeriesConcepts.pdf Modeling Financial Time Series with S-PLUS]. Second Edition. ([[PDF]])</ref>{{rp|p.60}} Other authors use strongly white and weakly white instead.<ref name="diebold">[[Francis X. Diebold]] (2007), ''[https://www.sas.upenn.edu/~fdiebold/Teaching221/FullBook.pdf Elements of Forecasting],'' 4th edition. ([[PDF]])</ref>

An example of a random vector that is Gaussian white noise in the weak but not in the strong sense is <math>x=[x_1,x_2]</math> where <math>x_1</math> is a normal random variable with zero mean, and <math>x_2</math> is equal to <math>+x_1</math> or to <math>-x_1</math>, with equal probability. These two variables are uncorrelated and individually normally distributed, but they are not jointly normally distributed and are not independent. If <math>x</math> 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 <math>w</math> to have non-zero expected value <math>\mu</math>. In [[image processing]] especially, where samples are typically restricted to positive values, one often takes <math>\mu</math> to be one half of the maximum sample value. In that case, the Fourier coefficient <math>W_0</math> corresponding to the zero-frequency component (essentially, the average of the <math>w_i</math>) will also have a non-zero expected value <math>\mu\sqrt{n}</math>; and the power spectrum <math>P</math> will be flat only over the non-zero frequencies.