# 諧波小波轉換

## 定義與性質

### 基礎推理

${\displaystyle W_{e}(\omega )={\begin{cases}{\frac {1}{4\pi }}&{\mbox{for }}-4\pi \leq \omega <-2\pi \\{\frac {1}{4\pi }}&{\mbox{for }}\quad 2\pi \leq \omega <4\pi \\0&{\mbox{elsewhere}}\end{cases}}}$

${\displaystyle w_{e}(x)=\int _{-\infty }^{\infty }W_{e}(\omega )e^{i\omega x}dw={\frac {sin4\pi x-sin2\pi x}{2\pi x}}}$

${\displaystyle W_{e}(\omega )={\begin{cases}{\frac {i}{4\pi }}&{\mbox{for }}-4\pi \leq \omega <-2\pi \\{\frac {-i}{4\pi }}&{\mbox{for }}2\pi \leq \omega <4\pi \\0&{\mbox{elsewhere}}\end{cases}}}$

${\displaystyle w_{o}(x)=\int _{-\infty }^{\infty }W_{o}(\omega )e^{i\omega x}dw={\frac {-(cos4\pi x-cos2\pi x)}{2\pi x}}}$

${\displaystyle w(x)={\frac {e^{i4\pi x}-e^{i2\pi x}}{i2\pi x}}}$

${\displaystyle W(\omega )=W_{e}(\omega )+iW_{o}(\omega )={\begin{cases}{\frac {1}{2\pi }}&{\mbox{for }}2\pi \leq \omega <4\pi \\0&{\mbox{elsewhere}}\end{cases}}}$

### 一系列的諧波小波

${\displaystyle w(x)\Rightarrow w(2^{j}x-k)={\frac {e^{i4\pi (2^{j}x-k)}-e^{i2\pi (2^{j}x-k)}}{i2\pi (2^{j}-k)}}=v(x)}$

${\displaystyle V(\omega )={\frac {1}{2^{j}}}e^{\frac {-i\omega k}{2^{j}}}W({\frac {\omega }{2^{j}}})}$

### 低頻頻帶（Zero-frequency band）

${\displaystyle \phi (x)={\frac {e^{i2\pi x}-1}{i2\pi x}}}$，其頻域特性將是一個介於${\displaystyle [0,2\pi )}$的方波，振幅為${\displaystyle {\frac {1}{2\pi }}}$

### 正交（Orthogonality）

${\displaystyle \int _{-\infty }^{\infty }w(x)v(x)dx=\int _{-\infty }^{infty}W(\omega )V(-\omega )dx}$

${\displaystyle \int _{-\infty }^{infty}w(x)v^{*}(x)dx=\int _{-\infty }^{infty}W(\omega )V^{*}(\omega )dx}$

${\displaystyle \int _{2\pi }^{4\pi }e^{i\omega k}d\omega =0}$

## 諧波小波轉換

${\displaystyle f(t)=\sum _{k=-\infty }^{\infty }\left[a_{k}\phi (t-k)+{\tilde {a}}_{k}\phi ^{*}(t-k)\right]+\sum _{j=0}^{\infty }\sum _{k=-\infty }^{\infty }\left[a_{j,k}w(2^{j}t-k)+{\tilde {a}}_{j,k}w^{*}(2^{j}t-k)\right].}$

{\displaystyle {\begin{aligned}a_{j,k}&{}=2^{j}\int _{-\infty }^{\infty }f(t)\cdot w^{*}(2^{j}t-k)\,dt\\{\tilde {a}}_{j,k}&{}=2^{j}\int _{-\infty }^{\infty }f(t)\cdot w(2^{j}t-k)\,dt\\a_{k}&{}=\int _{-\infty }^{\infty }f(t)\cdot \phi ^{*}(t-k)\,dt\\{\tilde {a}}_{k}&{}=\int _{-\infty }^{\infty }f(t)\cdot \phi (t-k)\,dt.\end{aligned}}}

## 參考資料

1. Newland, David. "Harmonic Wavelet Analysis". Proceedings of the Royal Society of London, Series A (Mathematical and Physical Sciences). Oct 1993, 443 (1917): 203 – 225.
2. Lokenath Debnath. Wavelet Transforms and Their Applications. Boston: Birkhäuser. 2002: 475–490. ISBN 0817642048.