# 希爾伯特轉換

（重定向自希尔伯特变换

## 定義

u 的希尔伯特变换可以认为是 u(t) 与函数 h(t) = 1/(πt) 的卷积。由于 h(t) 是不可积的英语Integrable system，定义卷积的积分不收敛。因而希尔伯特变换是使用柯西主值（这里记为p.v.）定义的。准确说来，函数（或信号） u(t) 的希尔伯特变换是：

${\displaystyle H(u)(t)=\operatorname {p.v.} \int _{-\infty }^{\infty }u(\tau )h(t-\tau )\,d\tau ={\frac {1}{\pi }}\ \operatorname {p.v.} \int _{-\infty }^{\infty }{\frac {u(\tau )}{t-\tau }}\,d\tau }$

${\displaystyle H(u)(t)=-{\frac {1}{\pi }}\lim _{\varepsilon \rightarrow 0}\int _{\varepsilon }^{\infty }{\frac {u(t+\tau )-u(t-\tau )}{\tau }}\,d\tau .}$

${\displaystyle H(H(u))(t)=-u(t)}$

### 頻率響應

${\displaystyle H(\omega )=(-i\cdot \operatorname {sgn}(\omega ))\cdot {\mathcal {F}}\{h\}(\omega )\,}$

• ${\displaystyle {\mathcal {F}}}$ 是傅立葉變換，
• i (有時寫作j )是虛數單位
• ${\displaystyle \omega \,}$ 角頻率，以及
• ${\displaystyle \operatorname {sgn}(\omega )={\begin{cases}\ \ 1,&{\mbox{for }}\omega >0,\\\ \ 0,&{\mbox{for }}\omega =0,\\-1,&{\mbox{for }}\omega <0,\end{cases}}}$

${\displaystyle {\mathcal {F}}\{{\widehat {s}}\}(\omega )=H(\omega )\cdot {\mathcal {F}}\{s\}(\omega )}$ ,

### 反（逆）希爾伯特轉換

${\displaystyle {\mathcal {F}}\{s\}(\omega )=-H(\omega )\cdot {\mathcal {F}}\{{\widehat {s}}\}(\omega )}$

${\displaystyle s(t)=-(h*{\widehat {s}})(t)=-{\mathcal {H}}\{{\widehat {s}}\}(t).\,}$

## 希爾伯特轉換表格

${\displaystyle u(t)\,}$

${\displaystyle H(u)(t)}$
${\displaystyle \sin(t)}$  [fn 2] ${\displaystyle -\cos(t)}$
${\displaystyle \cos(t)}$  [fn 2] ${\displaystyle \sin(t)\,}$
${\displaystyle \exp \left(it\right)}$  ${\displaystyle -i\exp \left(it\right)}$
${\displaystyle \exp \left(-it\right)}$  ${\displaystyle i\exp \left(-it\right)}$
${\displaystyle 1 \over t^{2}+1}$  ${\displaystyle t \over t^{2}+1}$
${\displaystyle e^{-t^{2}}}$  ${\displaystyle 2\pi ^{-1/2}F(t)}$  参见道森积分
Sinc函数
${\displaystyle \sin(t) \over t}$
${\displaystyle 1-\cos(t) \over t}$

${\displaystyle \sqcap (t)}$
${\displaystyle {1 \over \pi }\log \left|{t+{1 \over 2} \over t-{1 \over 2}}\right|}$

${\displaystyle \delta (t)\,}$
${\displaystyle {1 \over \pi t}}$

${\displaystyle \chi _{[a,b]}(t)\,}$
${\displaystyle {\frac {1}{\pi }}\log \left\vert {\frac {t-a}{t-b}}\right\vert }$
Notes
1. ^ Some authors (e.g., Bracewell) use our −H as their definition of the forward transform. A consequence is that the right column of this table would be negated.
2. The Hilbert transform of the sin and cos functions can be defined in a distributional sense, if there is a concern that the integral defining them is otherwise conditionally convergent. In the periodic setting this result holds without any difficulty.

## 特性

### 邊界

${\displaystyle \|Hu\|_{p}\leq C_{p}\|u\|_{p}}$

${\displaystyle C_{p}={\begin{cases}\tan {\frac {\pi }{2p}}&{\text{for }}1

${\displaystyle S_{R}f=\int _{-R}^{R}{\hat {f}}({\xi })e^{2\pi ix\xi }\,d\xi }$

### 反自伴性

${\displaystyle \langle Hu,v\rangle =\langle u,-Hv\rangle }$

u ∈ Lp(R) 且 v ∈ Lq(R) （Titchmarsh 1948，Theorem 102）.

### 逆轉換

${\displaystyle H(H(u))=-u}$

${\displaystyle H^{-1}=-H}$

### 微分

${\displaystyle H\left({\frac {du}{dt}}\right)={\frac {d}{dt}}H(u)}$

${\displaystyle H\left({\frac {d^{k}u}{dt^{k}}}\right)={\frac {d^{k}}{dt^{k}}}H(u)}$

### 旋積

${\displaystyle h(t)={\text{p.v. }}{\frac {1}{\pi t}}}$

${\displaystyle H(u)=h*u}$

${\displaystyle H(u)(t)={\frac {d}{dt}}\left({\frac {1}{\pi }}(u*\log |\cdot |)(t)\right)}$

${\displaystyle H(u*v)=H(u)*v=u*H(v)}$

uv 為緊支撐分布，則此項論述嚴格成立，在這個狀況下

${\displaystyle h*(u*v)=(h*u)*v=u*(h*v)}$

### 不變性

• 可與算子 Taƒ(x) = ƒ(x + a) 交換，對所有實數 a
• 可與算子 Mλƒ(x) = ƒ(λx) 交換，對所有 λ > 0
• 可與鏡射 Rƒ(x) = ƒ(−x) 反交換

${\displaystyle \displaystyle {U_{g}^{-1}f(x)=(cx+d)^{-1}f\left({ax+b \over cx+d}\right),\,\,\,g={\begin{pmatrix}a&b\\c&d\end{pmatrix}}}}$

## 離散希爾伯特轉換

${\displaystyle H(u)[n]=\scriptstyle {DTFT}^{-1}\displaystyle \{U(\omega )\cdot \sigma _{H}(\omega )\}}$

${\displaystyle \sigma _{H}(\omega )\ {\stackrel {\mathrm {def} }{=}}\ {\begin{cases}e^{+i\pi /2},&-\pi <\omega <0\\e^{-i\pi /2},&0<\omega <\pi \\0,&\omega =-\pi ,0,\pi \end{cases}}}$

${\displaystyle H(u)[n]=u[n]*h[n]}$

${\displaystyle h[n]\ {\stackrel {\mathrm {def} }{=}}\ \scriptstyle {DTFT}^{-1}{\big \{}\displaystyle \sigma _{H}(\omega ){\big \}}={\begin{cases}0,&{\mbox{for }}n{\mbox{ even}}\\{\frac {2}{\pi n}}&{\mbox{for }}n{\mbox{ odd}}\end{cases}}}$

${\displaystyle h_{N}[n]\ {\stackrel {\text{def}}{=}}\ \sum _{m=-\infty }^{\infty }h[n-mN]}$

MATLAB中有一函數 hilbert(u,N)，此函數會回傳一複數序列，其中虛部序列為 u[n]之離散希爾伯特轉換近似，實部序列為原本輸入之序列，所以這樣的複數輸出等於是 u[n]的分析訊號。與前述類似， hilbert(u, N) 只使用來自 sgn(ω)分佈的取樣，因此是與 hN[n] 的摺積。如前段所述，失真可藉由選擇比實際之u[n]序列更大的N與捨棄適當數量的輸出取樣來有效減少。圖 4為這種失真的一個例子。