諧波小波轉換

定義與性質

基礎推理

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

$w_e(x) = \int_{-\infty}^{\infty} W_e(\omega)e^{i\omega x} dw = \frac{sin4\pi x - sin2\pi x}{2\pi x}$

$W_e(\omega) = \begin{cases} \frac{i}{4\pi} & \mbox{for } -4\pi \le \omega < -2\pi \\ \frac{-i}{4\pi} & \mbox{for } 2\pi \le \omega < 4\pi \\ 0 & \mbox{elsewhere} \end{cases}$

$w_o(x) = \int_{-\infty}^{\infty} W_o(\omega)e^{i\omega x} dw = \frac{-(cos4\pi x - cos2\pi x)}{2\pi x}$

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

$W(\omega) = W_e(\omega) + iW_o(\omega) = \begin{cases} \frac{1}{2\pi} & \mbox{for } 2\pi \le \omega < 4\pi \\ 0 & \mbox{elsewhere} \end{cases}$

一系列的諧波小波

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

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

低頻頻帶（Zero-frequency band）

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

正交（Orthogonality）

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

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

$\int_{2\pi}^{4\pi} e^{i\omega k}d\omega = 0$

諧波小波轉換

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

\begin{align} 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{align}

參考資料

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