White noise vector A
random vector (that is, a random variable with values in
Rn) is said to be a white noise vector or white random vector if its components each have a
probability distribution with zero mean and a finite
variance. While standard signal processing definitions strictly require these variances to be identical to guarantee a perfectly flat power spectrum, broader statistical treatments sometimes only require the variances to be finite and the components to be
statistically independent: that is, their
joint probability distribution must be the product of the distributions of the individual components. A necessary (but,
in general, not sufficient) condition for statistical independence of two variables is that they be
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
Rii is the
variance of component
wi; and the
correlation matrix must be the
n by
n identity matrix. (If the variances are identical, the covariance matrix simplifies to a scalar multiple of the identity matrix, R = \sigma^2 I). If, in addition to being independent, every variable in
w also has a
normal distribution with zero mean and the same variance \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 \sigma^2. 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,
Pi = E(|
Wi|2). Under that definition, a Gaussian white noise vector will have a perfectly flat power spectrum, with
Pi =
σ2 for all
i. If
w is a white random vector, but not a Gaussian one, its Fourier coefficients
Wi 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. Other authors use strongly white and weakly white instead. An example of a random vector that is Gaussian white noise in the weak but not in the strong sense is x=[x_1,x_2] where x_1 is a normal random variable with zero mean, and x_2 is equal to +x_1 or to -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 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 w to have non-zero expected value \mu. In
image processing especially, where samples are typically restricted to positive values, one often takes \mu to be one half of the maximum sample value. In that case, the Fourier coefficient W_0 corresponding to the zero-frequency component (essentially, the average of the w_i) will also have a non-zero expected value \mu\sqrt{n}; and the power spectrum P will be flat only over the non-zero frequencies.
Discrete-time white noise A discrete-time
stochastic process W(n) is a generalization of a random vector with a finite number of components to infinitely many components. A discrete-time stochastic process W(n) is called
weak-sense white noise (or often simply "white noise" in signal processing) if its mean is equal to zero for all n, i.e., \operatorname{E}[W(n)] = 0, and if its
autocorrelation function R_{W}(n) = \operatorname{E}[W(k+n)W(k)] has a non-zero value only for n = 0, i.e., R_{W}(n) = \sigma^2 \delta(n), where \sigma^2 is the variance and \delta(n) is the
Kronecker delta. The variables W(n) are only required to be uncorrelated (not necessarily
statistically independent) because the characteristic flat
power spectral density of white noise depends entirely on the process's second-order moments (its autocorrelation function). A process that strictly requires the samples to be independent and identically distributed (i.i.d.) is referred to as
strict-sense white noise. Notably, if the uncorrelated random variables are jointly
Gaussian, this inherently guarantees they are also statistically independent.
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 w of a real-valued parameter t. Such a process is said to be white noise in the strongest sense if the value w(t) for any time t is a random variable that is statistically independent of its entire history before t. A weaker definition requires independence only between the values w(t_1) and w(t_2) at every pair of distinct times t_1 and t_2. An even weaker definition requires only that such pairs w(t_1) and w(t_2) be uncorrelated. 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 w is no longer a finite-dimensional space \mathbb{R}^n, but an infinite-dimensional
function space. Moreover, by any definition a white noise signal w would have to be essentially discontinuous at every point; therefore even the simplest operations on w, like integration over a finite interval, require advanced mathematical machinery. Some introductory engineering texts treat this as a heuristic rather than a strict mathematical definition, and require each value w(t) to be a real-valued random variable with expectation \mu and some finite variance \sigma^2. Then the covariance \mathrm{E}(w(t_1)\cdot w(t_2)) between the values at two times t_1 and t_2 is well-defined: it is zero if the times are distinct, and \sigma^2 if they are equal. However, by this definition, the integral : W_{[a,a+r]} = \int_a^{a+r} w(t)\, dt over any interval with positive width r would be simply the width times the expectation: r\mu. While the expectation being zero is not inherently problematic, if the pointwise variance is finite, the variance of this integral would be zero, meaning the signal has no measurable energy. This property renders the concept inadequate as a model of white noise signals either in a physical or mathematical sense, because a true white noise process must possess a flat
power spectral density, which requires the pointwise variance to be infinite rather than finite. Therefore, most authors define the signal w indirectly by specifying random values for the integrals of w(t) and |w(t)|^2 over each interval [a,a+r]. In this approach, however, the value of w(t) at an isolated time cannot be defined as a standard real-valued random variable. Also the covariance \mathrm{E}(w(t_1)\cdot w(t_2)) becomes infinite when t_1=t_2; and the
autocovariance function (often loosely referred to as autocorrelation in engineering contexts) \mathrm{R}(t_1,t_2) must be defined as N \delta(t_1-t_2), where N is some real constant representing the power spectral density and \delta is the
Dirac delta function, an unbounded measure which correctly reflects the infinite variance at t_1=t_2. In this approach, one usually specifies that the integral W_I of w(t) over an interval I=[a,b] is a real random variable with normal distribution, zero mean, and variance (b-a)\sigma^2; and also that the covariance \mathrm{E}(W_I\cdot W_J) of the integrals W_I, W_J is r\sigma^2, where r is the width of the intersection I\cap J of the two intervals 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 w is defined as a stochastic tempered distribution, i.e. a random variable with values in the space \mathcal S'(\mathbb R) of
tempered distributions. Analogous to the case for finite-dimensional random vectors, a probability law on the infinite-dimensional space \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 X \sim \mathcal N_n (\mu , \Sigma ), which has characteristic function: : \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 w : \Omega \to \mathcal S'(\mathbb R) must satisfy: : \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 \langle w, \varphi \rangle is the natural pairing of the tempered distribution w(\omega) with the Schwartz function \varphi (i.e. we consider \varphi as a fixed linear function on \mathcal S'(\mathbb R) analogous to k above). • And \| \varphi \|_2^2 = \int_{\mathbb R} \vert \varphi (x) \vert^2\,\mathrm d x . ==Mathematical applications==