The term white noise is also commonly applied to a noise signal in the spatial domain which has zero autocorrelation over the relevant space dimensions. The signal is then "white" in the spatial frequency domain (this is equally true for signals in the angular frequency domain, e.g. the distribution of a signal across all angles in the night sky). The image to the right displays a finite length, discrete time realization of a white noise process generated from a computer.

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 0 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 (i.e. noise with a Gaussian amplitude distribution - see normal distribution) is necessarily white noise. However, neither property implies the other. Gaussianity refers to the way signal values are distributed, while the term 'white' refers to correlations at two distinct times, which are independent of the noise amplitude distribution.

Pink noise (left) and white noise (right) on a spectral view

We can therefore find Gaussian white noise, but also Poisson, Cauchy, etc. white noises. Note that the distribution must have infinite variance. 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. These models are used so frequently that the term additive white Gaussian noise has a standard abbreviation: AWGN. Gaussian white noise has the useful statistical property that its values are independent (see Statistical independence).

White noise is the generalized mean-square derivative of the Wiener process or Brownian motion.

1. 若时间序列的自相关函数在k>3时都落入置 信 区间，且逐渐趋于零，则该时间序列具有平稳性.
2. 如果数据序列是非平稳的，并存在一定的增长或下降 趋势，则需要对数据进行差分处理.
3. 不稳定序列，必须对这两个变量进行适当的变换使之变为稳定序列，然后才可以对其进行格兰杰因果性检验.
4. 一般人们所关注的的有趋势和季节/循环成分的时间序列都不是平稳的。这时就需要对时间序列进行差分(difference)来消除这些不平稳的成分，而使其变成平稳的时间序列， 并估计ARMA模型，估计之后再转变该模型，使之适应于差分之前的序列这个过程和...
5. 平稳时间序列有三种重要的形式，即AR 序列、MA 序列、ARMA 序列。 非平稳序列. 方面，可以用ARIMA 序列来 ... 取0或1，一般不超过2可得到平稳ARMA(p,q) 序列[2],数据平稳化过后，可以用ARMA模型的参数估计方法. 对处理后的数据进行建模.

[此贴子已经被作者于2006-5-2 2:42:53编辑过]

In the mathematical sciences, a stationary process (or strict(ly) stationary process) is a stochastic process whose probability distribution at a fixed time or position is the same for all times or positions. As a result, parameters such as the mean and variance also do not change over time or position.

As an example, the measurement of white noise is stationary. Alternatively, the measurement of a cymbal clashing is not stationary. Although a cymbal clash is basically white noise, the measurement of that noise varies over time: Before the clash, there is silence, and after the clash, the noise gradually diminishes.

Stationarity is used as a tool in time series analysis, where the raw data are often transformed to become stationary, for example, economic data are often seasonal and/or dependent on the price level. Processes are described as trend stationary if they are a linear combination of a stationary process and one or more processes exhibiting a trend. Transforming this data to leave a stationary data set for analysis is referred to as de-trending.

A discrete-time stationary process where the sample space is also discrete (so that the random variable may take one of N possible values) is known as a Bernoulli scheme. When N = 2, the process is called a Bernoulli process.

[此贴子已经被作者于2006-5-2 2:55:16编辑过]

Weak or wide-sense stationarity

A weaker form of stationarity commonly employed in signal processing is known as weak-sense, wide-sense stationarity (WSS), second-order stationarity or covariance stationarity. WSS random processes only require that 1st and 2nd moments do not vary with respect to time.

So, a continuous-time random process x(t) which is WSS has the following restrictions on its mean function

1.

2.

The first property implies that the mean function mx(t) must be constant. The second property implies that the correlation function depends only on the difference between t1 and t2 and only needs to be indexed by one variable rather than two variables. Thus, instead of writing,