Editing White noise (section)

===Continuous-time white noise===
In order to define the notion of white noise in the theory of [[continuous-time]] 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 <math>w</math> of a real-valued parameter <math>t</math>.

Such a process is said to be white noise in the strongest sense if the value <math>w(t)</math> for any time <math>t</math> is a random variable that is statistically independent of its entire history before <math>t</math>. A weaker definition requires independence only between the values <math>w(t_1)</math> and <math>w(t_2)</math> at every pair of distinct times <math>t_1</math> and <math>t_2</math>. An even weaker definition requires only that such pairs <math>w(t_1)</math> and <math>w(t_2)</math> be uncorrelated.<ref name=econterms>[http://economics.about.com/od/economicsglossary/g/whitenoise.htm ''White noise process''] {{Webarchive|url=https://web.archive.org/web/20160911134507/http://economics.about.com/od/economicsglossary/g/whitenoise.htm |date=2016-09-11 }}. By Econterms via About.com. Accessed on 2013-02-12.</ref>  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 <math>w</math> is no longer a finite-dimensional space <math>\mathbb{R}^n</math>, but an infinite-dimensional [[function space]]. Moreover, by any definition a white noise signal <math>w</math> would have to be essentially discontinuous at every point; therefore even the simplest operations on <math>w</math>, like integration over a finite interval, require advanced mathematical machinery.

Some authors{{citation needed|date=October 2023}}{{what|reason=Since the definition proposed in this section is not remotely workable in a mathematical sense, I doubt that any authors do this. Instead, we are looking at a heuristic only.|date=October 2023}} require each value <math>w(t)</math> to be a real-valued random variable with expectation <math>\mu</math> and some finite variance <math>\sigma^2</math>. Then the covariance <math>\mathrm{E}(w(t_1)\cdot w(t_2))</math> between the values at two times <math>t_1</math> and <math>t_2</math> is well-defined: it is zero if the times are distinct, and <math>\sigma^2</math> if they are equal. However, by this definition, the integral
: <math>W_{[a,a+r]} = \int_a^{a+r} w(t)\, dt</math>
over any interval with positive width <math>r</math> would be simply the width times the expectation: <math>r\mu</math>.{{what|reason=The *expectation value* of the mean is zero. And this is not a problem.|date=October 2023}} This property renders the concept inadequate as a model of white noise signals either in a physical or mathematical sense.{{what|reason=Why?|date=October 2023}}

Therefore, most authors define the signal <math>w</math> indirectly by specifying random values for the integrals of <math>w(t)</math> and <math>|w(t)|^2</math> over each interval <math>[a,a+r]</math>. In this approach, however, the value of <math>w(t)</math> at an isolated time cannot be defined as a real-valued random variable{{Citation needed|reason=an authoritative work on white noise given one such example should be given|date=January 2017}}. Also the covariance <math>\mathrm{E}(w(t_1)\cdot w(t_2))</math> becomes infinite when <math>t_1=t_2</math>; and the [[autocorrelation]] function <math>\mathrm{R}(t_1,t_2)</math> must be defined as <math>N \delta(t_1-t_2)</math>, where <math>N</math> is some real constant and <math>\delta</math> is the [[Dirac delta function]].{{what|reason=Correlation can only take on values in [0,1], so N must be 1 and delta must take the value 1 for t_1 = t_2; it is not the dirac measure here. However, all these concepts are fishy.|date=October 2023}}

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

In the mathematical field known as [[white noise analysis]], a Gaussian white noise <math>w</math> is defined as a stochastic tempered distribution, i.e. a random variable with values in the space <math>\mathcal S'(\mathbb R)</math> of [[Distribution (mathematics)#Tempered distribution|tempered distributions]]. Analogous to the case for finite-dimensional random vectors, a probability law on the infinite-dimensional space <math>\mathcal S'(\mathbb R)</math> 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 <math>X \sim \mathcal N_n (\mu , \Sigma )</math>, which has characteristic function
: <math>\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 } ,</math>
the white noise <math>w : \Omega \to \mathcal S'(\mathbb R)</math> must satisfy
: <math>\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},</math>
where <math>\langle w, \varphi \rangle</math> is the natural pairing of the tempered distribution <math>w(\omega)</math> with the Schwartz function <math>\varphi</math>, taken scenariowise for <math>\omega \in \Omega</math>, and <math>\| \varphi \|_2^2 = \int_{\mathbb R} \vert \varphi (x) \vert^2\,\mathrm d x </math>.