# 白雜訊

（重定向自白噪音

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## 統計特性

${\displaystyle \mu _{n}=\mathbb {E} \{n(t)\}=0}$

${\displaystyle r_{nn}=\mathbb {E} \{n(t)n(t-\tau )\}=\delta (\tau )}$

## 數學定義

### 白色隨機向量

${\displaystyle \mu _{w}=\mathbb {E} \{\mathbf {w} \}=0}$
${\displaystyle R_{ww}=\mathbb {E} \{\mathbf {w} \mathbf {w} ^{T}\}=\sigma ^{2}\mathbf {I} }$

### 白色隨機過程(白雜訊)

${\displaystyle \mu _{w}(t)=\mathbb {E} \{w(t)\}=0}$
${\displaystyle R_{ww}(t_{1},t_{2})=\mathbb {E} \{w(t_{1})w(t_{2})\}=(N_{0}/2)\delta (t_{1}-t_{2})}$

${\displaystyle S_{xx}(\omega )=(N_{0}/2)\,\!}$

## 随机向量变换

### 模拟随机向量

${\displaystyle \,\!K_{xx}=E\Lambda E^{T}}$

${\displaystyle \mathbf {x} =H\,\mathbf {w} +\mu }$

${\displaystyle \,\!H=E\Lambda ^{1/2}}$

${\displaystyle \mathbb {E} \{\mathbf {x} \}=H\,\mathbb {E} \{\mathbf {w} \}+\mu =\mu }$

${\displaystyle \mathbb {E} \{(\mathbf {x} -\mu )(\mathbf {x} -\mu )^{T}\}=H\,\mathbb {E} \{\mathbf {w} \mathbf {w} ^{T}\}\,H^{T}=H\,H^{T}=E\Lambda ^{1/2}\Lambda ^{1/2}E^{T}=K_{xx}}$

### Whitening 随机向量

whitening 一个平均值${\displaystyle \mathbf {\mu } }$协方差矩阵${\displaystyle K_{xx}}$ 的向量 ${\displaystyle \mathbf {x} }$ 的方法是执行下面的计算：

${\displaystyle \mathbf {w} =\Lambda ^{-1/2}\,E^{T}\,(\mathbf {x} -\mathbf {\mu } )}$

${\displaystyle \mathbb {E} \{\mathbf {w} \}=\Lambda ^{-1/2}\,E^{T}\,(\mathbb {E} \{\mathbf {x} \}-\mathbf {\mu } )=\Lambda ^{-1/2}\,E^{T}\,(\mu -\mu )=0}$

${\displaystyle \mathbb {E} \{\mathbf {w} \mathbf {w} ^{T}\}=\mathbb {E} \{\Lambda ^{-1/2}\,E^{T}\,(\mathbf {x} -\mathbf {\mu } )(\mathbf {x} -\mathbf {\mu } )^{T}E\,\Lambda ^{-1/2}\,\}}$
${\displaystyle =\Lambda ^{-1/2}\,E^{T}\,\mathbb {E} \{(\mathbf {x} -\mathbf {\mu } )(\mathbf {x} -\mathbf {\mu } )^{T}\}E\,\Lambda ^{-1/2}\,}$
${\displaystyle =\Lambda ^{-1/2}\,E^{T}\,K_{xx}E\,\Lambda ^{-1/2}}$

${\displaystyle \Lambda ^{-1/2}\,E^{T}\,E\Lambda E^{T}E\,\Lambda ^{-1/2}=\Lambda ^{-1/2}\,\Lambda \,\Lambda ^{-1/2}=I}$

## 随机信号变换

### 模拟连续时间随机信号

${\displaystyle K_{x}(\tau )=\mathbb {E} \left\{(x(t_{1})-\mu )(x(t_{2})-\mu )^{*}\right\}{\mbox{ where }}\tau =t_{1}-t_{2}}$
${\displaystyle S_{x}(\omega )=\int _{-\infty }^{\infty }K_{x}(\tau )\,e^{-j\omega \tau }\,d\tau }$

${\displaystyle \int _{-\infty }^{\infty }{\frac {\log(S_{x}(\omega ))}{1+\omega ^{2}}}\,d\omega <\infty }$

${\displaystyle S_{x}(\omega )=|H(\omega )|^{2}=H(\omega )\,H^{*}(\omega )}$

${\displaystyle S_{x}(\omega )={\frac {\Pi _{k=1}^{N}(c_{k}-j\omega )(c_{k}^{*}+j\omega )}{\Pi _{k=1}^{D}(d_{k}-j\omega )(d_{k}^{*}+j\omega )}}}$

${\displaystyle {\hat {x}}(t)={\mathcal {F}}^{-1}\left\{H(\omega )\right\}*w(t)+\mu }$

${\displaystyle \mathbb {E} \{w(t)\}=0}$
${\displaystyle \mathbb {E} \{w(t_{1})w^{*}(t_{2})\}=K_{w}(t_{1},t_{2})=\delta (t_{1}-t_{2})}$

### 连续时间随机信号的白化

${\displaystyle H_{inv}(\omega )={\frac {1}{H(\omega )}}}$

${\displaystyle w(t)={\mathcal {F}}^{-1}\left\{H_{inv}(\omega )\right\}*(x(t)-\mu )}$

${\displaystyle S_{w}(\omega )={\mathcal {F}}\left\{\mathbb {E} \{w(t_{1})w(t_{2})\}\right\}=H_{inv}(\omega )S_{x}(\omega )H_{inv}^{*}(\omega )={\frac {S_{x}(\omega )}{S_{x}(\omega )}}=1}$

${\displaystyle K_{w}(\tau )=\,\!\delta (\tau )}$