Detailed understanding of white noise

In data processing , We often encounter noise . General noise is a kind of random signal or random process with constant power spectral density , That is white noise . The origin of white noise is that white light is a mixture of monochromatic light of various frequencies , Because the average power spectrum property of this signal becomes “ white ”, This signal is also known as white noise . The ideal white noise has infinite bandwidth , So its energy is infinite , This is impossible in the real world . actually , People often regard flat signals with limited bandwidth as white noise , To facilitate mathematical analysis . White noise has the following characteristics ：

• The mathematical expectation is 0：
  $${\mu }_n=E\{n(t)\}=0$$

#  Code testing ：
import numpy as np
noise = np.random.rand(0, 1, 100000)
mean = np.mean(noise)
print(mean)

0.004019681758208514  #  Why not 0, That is to say , As long as the bandwidth is enough , Is the ideal white noise .

• The autocorrelation function is Dirac function
  $$r_{nn}=E\{n(t)n(t-\tau )\}=\delta (\tau )$$

#  Code testing 
import imageio
import numpy as np

nums = np.arange(10, 100000, 1000)
for num in nums:
	noise = np.random.normal(0, 1, num)
	corr = np.correlate(noise, noise, mode='full')
	plt.plot(corr)
	plt.savefig('figs/fig%d.png'%num)
	plt.close()

with imageio.get_write('mygif.gif', mode='I') as writer:
	for num in nums:
		image = imageio.imread('figs/fig%d/png%num)
		write.append_data(image)

Image results ：
Above is Dirac function .

• Power spectral density is flat

import matplotlib.pyplot as plt
noise = np.random.rand(0, 1, 100000)
plt.psd(noise, 1000)
plt.savefig('image.png', dpi=300)

give the result as follows ：