Gaussian noise, Gaussian white noise

Gaussian noise, Gaussian white noise, Matlab, WGN

[Original article source ]:

Http://www.cnblogs.com/YoungHit/archive/2012/03/09/2388230.html

This article describes white Gaussian noise (WGN ).

Baidu encyclopedia interprets it as "Gaussian white noise, the amplitude distribution follows the Gaussian distribution, and the power spectrum density follows the even distribution", which sounds obscure. The following is a common and detailed example.

White noise, like white light, is the superposition of all colors of light. The essential difference between the light of different colors is that their frequencies are different (for example, the red light is long and the frequency is low, corresponding, the wavelength is short and the frequency is high ). White Noise tends to be a constant value in the power spectrum (if the frequency is the horizontal axis and the square of the signal amplitude is the power), that is, the noise frequency is rich and has components in the entire spectrum, that is, from the low frequency to the high frequency, the low frequency refers to the constant or slow change of the signal, and the high frequency refers to the sudden change of the signal.

We can see from Fourier transform that the time domain is limited, the frequency domain is infinite, the frequency domain is limited, and the time domain is infinite. Then the infinite signal in the frequency domain is converted to the time domain, which corresponds to an integer multiple of the impact function (which can also be deduced by the formula :). That is to say, at a certain point in the timeline, the noise is isolated and irrelevant to the noise of other points. That is to say, the noise amplitude of this point can be arbitrary and is not affected by the noise amplitude of the front and back points. In short, the noise amplitude at any time point is random (in this sentence, the power spectral density is subject to the same meaning as the distribution. The difference is that the former is described from the time domain perspective, the latter is described in the frequency field ). The power spectrum density (
Spectral density, PSD), which defines how the signal power is distributed with frequency from the frequency domain perspective, that is, the frequency is the horizontal axis, and the power is the vertical axis.

Since the white noise signal is "random", what is "correlation" in turn? As the name suggests, the correlation is that the noise points at a certain time point are not isolated and are related to the noise amplitude at other times. In fact, there are many related situations, such as the noise amplitude at this moment is greater than that at the previous moment, and the noise amplitude at the next moment is greater than that at this moment, that is, the signal amplitude is arranged in ascending order on the timeline. In addition, the amplitude ranges from large to small, or the amplitude ranges from small to small are called "correlation", rather than "random.

After explaining "white noise", let's talk about "Gaussian distribution ". Gaussian distribution, also known as normal distribution ). The shape of the probability density function curve is determined by two parameters: Average Value and variance. In short, the mean value determines the symmetric midline of the curve, and the variance determines the fatness of the curve, that is, the degree close to the midline. Probability Density defines how the frequency of a signal changes with its amplitude, that is, the signal amplitude is the horizontal axis, and the frequency of occurrence is the vertical axis. Therefore, from the perspective of probability density, the amplitude distribution of Gaussian white noise is subject to Gaussian distribution.

The meanings of "White Noise" and "Gaussian noise" are described. Back to the beginning of the article, we will explain that the Gaussian white noise follows the Gaussian distribution, and the power spectral density follows the even distribution. Its meaning is very clear. The first half describes Gaussian noise from the perspective of airspace (amplitude), while the second half describes "White Noise" in the frequency field ".

The following uses the MATLAB program to demonstrate how to perceive Gaussian white noise.

 

Program 1 (Gaussian white noise ):

It can be seen that the power spectral density of Gaussian white noise is uniformly distributed.

If the noise is sorted from small to large, it is changed from random noise to correlated noise, and the power spectral density is no longer even.

Program 2 (Non-Gaussian white noise ):

Let's take a look at the characteristics of Gaussian white noise from the statistical information and amplitude distribution.

Program 3 (Gaussian white noise ):

The vertical axis of the histogram is frequency, while the vertical axis of probability density is frequency, but the approximate distribution curves of the two are the same. Therefore, this figure shows that the amplitude distribution of Gaussian white noise is subject to Gaussian distribution.