# 加性高斯白噪声

## 信道容量

AWGN 信道由一系列的${\displaystyle Y_{i}}$ （输出） 来表示，其中的 ${\displaystyle i}$  表示离散的时间事件索引。${\displaystyle Y_{i}}$ ${\displaystyle X_{i}}$ （输入）和${\displaystyle Z_{i}}$ （噪声）的数值和，其中 ${\displaystyle Z_{i}}$  是独立恒等分布的随机变量并来自于均值为 0，方差为 ${\displaystyle N}$ （噪声） 的正态分布。${\displaystyle Z_{i}}$  可以进一步认为和 ${\displaystyle X_{i}}$  有关。

${\displaystyle Z_{i}\sim {\mathcal {N}}(0,N)\,\!}$
${\displaystyle Y_{i}=X_{i}+Z_{i}\,\!}$

${\displaystyle {\frac {1}{k}}\sum _{i=1}^{k}x_{i}^{2}\leq P,\,\!}$

${\displaystyle C=\max _{f(x){\text{ s.t. }}E\left(X^{2}\right)\leq P}I(X;Y)\,\!}$

{\displaystyle {\begin{aligned}I(X;Y)=h(Y)-h(Y|X)&=h(Y)-h(X+Z|X)&=h(Y)-h(Z|X)\end{aligned}}\,\!}

${\displaystyle I(X;Y)=h(Y)-h(Z)\,\!}$

${\displaystyle h(Z)={\frac {1}{2}}\log(2\pi eN)\,\!}$

${\displaystyle E(Y^{2})=E(X+Z)^{2}=E(X^{2})+2E(X)E(Z)+E(Z^{2})=P+N\,\!}$

${\displaystyle h(Y)\leq {\frac {1}{2}}\log(2\pi e(P+N))\,\!}$

${\displaystyle I(X;Y)\leq {\frac {1}{2}}\log(2\pi e(P+N))-{\frac {1}{2}}\log(2\pi eN)\,\!}$

${\displaystyle X\sim {\mathcal {N}}(0,P)\,\!}$

${\displaystyle C={\frac {1}{2}}\log \left(1+{\frac {P}{N}}\right)\,\!}$

## 时域中的影响

${\displaystyle {\frac {\mathrm {positive\ zero\ crossings} }{\mathrm {second} }}={\frac {\mathrm {negative\ zero\ crossings} }{\mathrm {second} }}}$
${\displaystyle =f_{0}{\sqrt {\frac {\mathrm {SNR} +1+{\frac {B^{2}}{12f_{0}^{2}}}}{\mathrm {SNR} +1}}}}$

• f0 = 滤波的中心频率
• B = 滤波器带宽
• SNR = 线性关系上的信噪功率比