# 希爾伯特轉換

## 希爾伯特轉換表格

$u(t)\,$

$H(u)(t)$
$\sin(t)$ $-\cos(t)$
$\cos(t)$ $\sin(t)\,$
$\exp \left( i t \right)$ $- i \exp \left( i t \right)$
$\exp \left( -i t \right)$ $i \exp \left( -i t \right)$
$1 \over t^2 + 1$ $t \over t^2 + 1$
Sinc函数
$\sin(t) \over t$
$1 - \cos(t)\over t$

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

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

$\chi_{[a,b]}(t) \,$
$\frac{1}{\pi}\log \left \vert \frac{t - a}{t - b}\right \vert$
Notes

## 定義

$\widehat s(t) = \mathcal{H}\{s\} = h(t)*s(t) = \int_{-\infty}^{\infty} s(\tau) h(t-\tau) d\tau = \frac{1}{\pi}\int_{-\infty}^{\infty}\frac{s(\tau)}{t-\tau}\, d\tau.\,$

$h(t) = \frac{1}{\pi t}\,$

$s\in L^p(\mathbb{R})$，則$\mathcal{H}(s)$可被定義，且屬於$L^p(\mathbb{R})$；其中$1

### 頻率響應

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

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

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

### 反（逆）希爾伯特轉換

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

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

## 特性

### 邊界

$\|Hu\|_p \le C_p\| u\|_p$

$C_p = \begin{cases} \tan \frac{\pi}{2p} & \text{for } 1 < p \leq 2\\ \cot \frac{\pi}{2p} & \text{for } 2 < p < \infty \end{cases}$

$S_R f = \int_{-R}^{R}\hat{f}({\xi})e^{2\pi i x\xi}\,d\xi$

### 反自伴性

$\langle Hu, v \rangle = \langle u, -Hv \rangle$

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

### 逆轉換

$H(H(u)) = -u$

$H^{-1} = -H$

### 微分

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

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

### 旋積

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

$H(u) = h*u$

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

$H(u*v) = H(u)*v = u*H(v)$

uv 為緊支撐分布，則此項論述嚴格成立，在這個狀況下

$h*(u*v) = (h*u)*v = u*(h*v)$

### 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

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

## 離散希爾伯特轉換

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

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

$H(u)[n] = u[n] * h[n]$

$h[n]\ \stackrel{\mathrm{def}}{=}\ \scriptstyle{DTFT}^{-1} \big \{\displaystyle \sigma_H(\omega)\big \} = \begin{cases} 0, & \mbox{for }n\mbox{ even}\\ \frac2{\pi n} & \mbox{for }n\mbox{ odd} \end{cases}$

$h_N[n]\ \stackrel{\text{def}}{=}\ \sum_{m=-\infty}^{\infty} h[n - mN]$

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。

## 參考文獻

• Bracewell, R. The Fourier Transform and Its Applications 2nd ed. McGraw-Hill. 1986.
• Carlson, Crilly, and Rutledge. Communication Systems 4th ed. 2002.
• Grafakos, Loukas, Classical and Modern Fourier Analysis, Pearson Education, Inc., 253–257, 2004, ISBN 0-13-035399-X.
• Pichorides, S., On the best value of the constants in the theorems of Riesz, Zygmund, and Kolmogorov, Studia Mathematica, 1972, 44: 165–179
• Riesz, Marcel, Sur les fonctions conjuguées, Mathematische Zeitschrift, 1928, 27 (1): 218–244, doi:10.1007/BF01171098
• Duoandikoetxea, J., Fourier Analysis, American Mathematical Society, 2000, ISBN 0-8218-2172-5
• Titchmarsh, E, Reciprocal formulae involving series and integrals, Mathematische Zeitschrift, 1926, 25 (1): 321–347, doi:10.1007/BF01283842.
• Titchmarsh, E, Introduction to the theory of Fourier integrals 2nd, Oxford University: Clarendon Press, 1948 (1986), ISBN 978-0-8284-0324-5.
• Pandey, J.N., The Hilbert transform of Schwartz distributions and applications, Wiley-Interscience, 1996, ISBN 0-471-03373-1
• Duistermaat, J.J., Distributions, Birkhäuser, 2010, doi:10.1007/978-0-8176-4675-2, ISBN 978-0-8176-4672-1