White Gaussian noise (WGN)

In this paper, Gaussian white noise （white Gaussian noise,WGN）.

　　 Baidu Encyclopedia explained as “ white Gaussian noise , The amplitude distribution obeys Gaussian distribution , The power spectral density follows a uniform distribution ”, It sounds a little obscure , The following is a simple and detailed introduction to .

　　 White noise , Like white light , It is the superposition of all colors of light , The essential difference between different colors of light is that their frequencies are different （ For example, the red light wave is long and the frequency is low , Corresponding , Purple light wave is long and short with high frequency ）. White noise in the power spectrum （ If you take frequency as the horizontal axis , The square of the signal amplitude is power ） The approach is constant , That is, the noise frequency is rich , There are components in the whole spectrum , That is, from low frequency to high frequency , Low frequency refers to the constant or slow change of the signal , High frequency refers to signal mutation .

　　 From the properties of Fourier transform , Finite time domain , The frequency domain is infinite ; Finite frequency domain , The time domain is infinite . Then the infinite signal in the frequency domain is transformed into the time domain , Corresponding to the integral multiple of the impact function （ It can also be deduced from the formula ：）. That is, at a certain point in the timeline , Noise isolation , It has nothing to do with the noise at other points , in other words , The noise amplitude at this point can be arbitrary , It is not affected by the noise amplitude of the front and rear points . In short , The noise amplitude at any time is random （ This sentence actually means The power spectral density follows the uniform distribution It means , The difference is , The former describes... From the perspective of time domain , The latter is described from the perspective of frequency domain ）. Here it should be pointed out Power spectral density （Power Spectral Density,PSD） The concept of , It starts from the frequency domain , It defines how the power of the signal is distributed with frequency , That is, taking the frequency as the horizontal axis , Power is the vertical axis .

　　 Since the white noise signal is “ Random ” Of , So in turn , What is called “ relevant ” Well ？ seeing the name of a thing one thinks of its function , Correlation is that the noise point at a certain time is not isolated , It is related to the noise amplitude at other times . In fact, there are many related situations , The noise amplitude at this time is larger than that at the previous time , The noise amplitude at the next moment is larger than that at this moment , That is, the amplitude of the signal is arranged in the order from small to large on the time axis . besides , Amplitude from large to small , Or amplitude one big one small and so on are called “ relevant ”, Instead of “ Random ” Of .

　　 The explanation is over “ White noise ”, Let 's talk about it again “ Gaussian distribution ”. Gaussian distribution , Also known as normal distribution （normal distribution）. The shape of the probability density function curve is determined by two parameters ： Mean and variance . Simply speaking , The mean value determines the symmetrical center line of the curve , The variance determines the weight of the curve , That is, the degree of being close to the middle line . Probability density It defines how the frequency of the signal changes with its amplitude , That is, take the signal amplitude as the horizontal axis , Take the frequency of occurrence as the longitudinal axis . therefore , In terms of probability density , Gaussian white noise The amplitude distribution obeys Gaussian distribution

　　 It describes “ White noise ” and “ Gaussian noise ” Two meanings , that , Go back to the explanation at the beginning of the article ： white Gaussian noise , The amplitude distribution obeys Gaussian distribution , The power spectral density follows a uniform distribution . Its meaning is very clear , The first half of the sentence is from the airspace ( amplitude ) Angle description “ Gaussian noise ”, The second half of the sentence describes... From the perspective of frequency domain “ White noise ”.

　　 Let's say matlab Program demonstration , Get to know Gaussian white noise .

Program 1（ white Gaussian noise ）：

As can be seen from the figure above , Gaussian white noise The power spectral density follows a uniform distribution .

If the noise is sorted from small to large , Then it changes from random noise to correlated noise , Then the power spectral density is no longer uniformly distributed .

Program 2（ Non Gaussian white noise ）：

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

Program 3（ white Gaussian noise ）：

The vertical axis of the histogram is frequency , And the vertical axis of probability density is frequency , But the approximate distribution curve of the two is the same , therefore , This picture explains the effect of Gaussian white noise The amplitude distribution obeys Gaussian distribution .