高斯噪声和白噪声

(1.2.69)
Phys. Meaning: The N Gaussian variables will be statistical each other, if
物理含义: 如果N个高斯随机变量之间是互不相关的,则它们 之间也是统计独立的。
4、满足高斯分布的充分条件：
The sufficient & necessary condition for RV to obey Gaussian distribution
(1.2.67)
where M is the matrix of the joint 2-order center moment （联合二阶中心矩） of the RV, M is its determinant (行列式), of the element
M ik is the surplus factor (余因子）
• 单（多）脉冲噪声：瞬态分析法
Single (multiplex) pulse noises: instantaneous analysis
一、高斯噪声（依噪声幅度分布特性判定）
Gaussian Noise: Judged according to the magnitude distribution feature
The linear combination of Gaussian noise is still a Gaussian noise.
<2> 高斯噪声与一固定数值相加的结果只改变噪声平均值,不 改变其它特性 The results of a Gaussian noise plus a fixed value
（2）性质: 由纯正弦单色光波或宽带热辐射光束产生的光子计数， 服从泊松分布。
Feature: The photon count generated by a pure monochromatic sine wave or wideband thermal radiation obeys Poisson Distribution.
5、高斯分布的特点与高斯噪声特性：The Features of Gaussian
Distribution & the Properties of Gaussian Noises
（1）高斯分布的特点：The features of Gaussian Distribution <1> 以 x = m 为轴，呈对称分布， x = m 时取最大值。 The distribution is symmetrical taking x = m as the axis of symmetry,
µik.
µ11 µ M = 21 .. µN1
µ12 µ22
..
... .... ..
µN2 ....
.. µNN
µ1N µ2N
and
µik = µki =< (xi − < xi >)(xk − < xk >) >
(1.2.68)
when the xi s are incorrelated each other, we have µik = 0 and M ik = 0 for i ≠ k . In this case, Eq. (1.2.67) will change into following form
（1）客观背景：The objective background 事实上，噪声函数的瞬时值可视为大量的相互独立的被加 项之和,且任意一个被加项与其它被加项相比,在方差或功率上 都相差无几。
In fact, a noise function can be regarded as the sum of great number of statistically independent summands (被加数）, and any of the summand has no obvious difference in variance or power compared with the others.
中心矩: The center moment:
µN =
1
∞
) −(x−m 2 2 2 σ
π 2 σ
−∞
)N ∫ (x −m e
dx
(1.2.66-1)
原点矩: The origin moment:
µn =
1
∞
−x2
2
0 = N σ
2 σ −∞ π
xNe2σ dx ∫
N 奇 为 数 N 偶 为 数
N (x − < x > 2 ) i i p(x) = ex − ∑ p 2 N N 2 i σ i= 1 2 (2 ) π σ ∏ i 1
i= 1
= p(x ) p(x2 ) p(xN ) ... 1
they are incorrelated each other.
3、高斯分布律：The Gaussian Distribution Law （1）一维概率密度函数: The 1-D probability density function 是由均值 m 和均方差
σ2
唯一确定的函数。
The 1-D PDF is uniquely determined by mean value
2.3
高斯噪声和白噪声
Gaussian Noise & White Noise
引言: 噪声分析的两类方法：The two kinds of methods for noise analysis
• 随机噪声：服从统计规律，用随机函数描述
Random noises: obey statistical law, described with random functions
2 x −(x−m)
2 σ −∞ π
−x2
∫e
2σ 2 dx
(1.2.64)
<3> 当 m= 0 时, when the m is zero, the PDF is ,
p x) = ( 1 2 σ π e
2 2 σ
(1.2.65)
<4> 高斯变量X的 n 阶中心矩与 n 阶原点矩
The n-order center moment & n-order origin moment of Gaussian variables
and it will reach its maximum value when
x =m .
<2> x →±∞ 时逼近横轴 It will approach to the horizontal axis
when x →±∞ .
<3> x = ± 处有拐点 The distribution curve has two inflection σ points at x = ± , respectively. σ
（3）结果：The result 此时，个别分量在很宽范围内的分布特性无关紧要。
In this situation, the distribution property of individual component will have no influence to that formed by the total, in a wide range.
（3）泊松分布: Poisson Distribution <1> 概率密度：Probability density function
<4> m−3σ < x < m+3σ 域内的概率为99.7%
The probability of that x falls in the range of m−3 < x < m+3 is 99.7%. σ σ
m− 2 < x < m+ 2 域内的概率为95.4% σ σ
The probability of that x falls in the range of m− 2 < x < m+ 2 is 95.4%. σ σ
m & variance σ 2 .
<1> 概率密度：The probability density function
p(x) = 1 2 σ π
) −(x−m 2
e
2 2 σ
(1.2.63)
<2> 分布函数: The distribution function
F(x) = P(X < x) = 1
m−σ < x < m+σ 域内的概率为68.3%
The probability of that x falls in the range of
m−σ < x < m+σ is 68.3%.
（2）高斯噪声特性：The features of Gaussian noises <1> 高斯噪声的线性组合仍是高斯噪声
Is that only the mean value is changed but not the others.
<3> 对独立的噪声源产生的噪声求和时, 按功率相加
The sum of the noises generated by independent sources is obtained by the adding of the powers of these noises.
(1.2.66-2)
（2）高斯分布的N维联合概率密度
The N-D joint probability density function of the Gaussian Distribution
p(x) = p(x , x2,..., xN ) 1
= 1 (2 ) π
N 2
M
1 2
1 N N ex − p ∑ ∑ M ik (xi − < xi >) ⋅ (xk − < xk >) 2 M i=1k=1
i= 1 N
律在
N →∞
的极限情况下趋于高斯分布律。
N
The central limit theorem (Liapunov Theorem): The distribution of the sum Z = ∑xi of a great number N of statistical
i= 1
independent random variables, with limit mathematical expectations and variances will approach to the Gaussian distribution under the conditio;4> 高斯噪声通过线性系统后, 仍是高斯噪声
It will still be Gaussian noises after their passing through linear systems.
6、光子噪声 Photon Noises （1）定义: 光辐射强度随机起伏导致的探测器输出信号噪声。
Def. : The output noise of a detector generated by the random fluctuation of the incident optical intensity is defined as the photon noise.
1、定义：幅度起伏遵从高斯分布的噪声
Def. : The noises, whose magnitude obeys Gaussian Distribution are defined as the Gaussian Noises.
2、中心极限定理（李雅普诺夫定理）：大量N个统计独立的、 具有有限的数学期望和方差的随机变量之和 Z = ∑xi 的分布
（2）满足高斯分布的条件：
The condition should be fulfilled for the RVs obeying Gaussian Distribution:
当被加项的数目很大而每一个被加项与所有被加项的总贡 献比很小时，这些随机变量之和的分布即趋于高斯分布。
The sum of RVs’ statistical distribution will approach to Gauss.Distrib. when the number of the summands is very large and the contribution of each summand is very little comparing with that of the total.