# 加伯–韋格納轉換

（重定向自加伯 韋格納轉換

## 數理定義

• 加伯 韋格納轉換

1. $D_x(t,f)=G_x(t,f)\times W_x(t,f)$

1.  :$G_x(t,f) = \int_{-\infty}^\infty e^{-\pi(\tau-t)^2}e^{-j2\pi f\tau}x(\tau) \, d\tau$
2.  :$W_x(t,f)=\int_{-\infty}^\infty x(t+\tau/2)x^*(t-\tau/2)e^{-j2\pi\tau\,f} \, d\tau$

1. $D_x(t,f)=\min\left\{|G_x(t,f)|^2,|W_x(t,f)|\right\}$
2. $D_x(t,f)=W_x(t,f)\times \{|G_x(t,f)|>0.25\}$
3. $D_x(t,f)=G_x^{2.6}(t,f)W_x^{0.7}(t,f)$

## 特性

### 不會有cross-term問題

cross-term問題主要發生在韋格納轉換($W_x(t,f)$)的過程中，因韋格納轉換並非線性，當被轉換函式x(t)有超過兩個物件(component)或其因次(order)超過三，就有可能在時間-頻率關係圖中產生干擾(distortion)，導致Cross-Talk的產生。 考慮函式$x(t)=ag(t)+bs(t)$ 根據定義:

    $W_x(t,f)=\int_{-\infty}^\infty x(t+\tau/2)x^*(t-\tau/2)e^{-j2\pi\tau\,f} \, d\tau$


    $W_x(t,f)=\int_{-\infty}^\infty [ag(t+\tau/2)+bs(t+\tau/2)] [a^*g(t-\tau/2)+b^*s(t+\tau/2)]e^{-j2\pi\tau\,f} \, d\tau$
$W_x(t,f)=|a^2|*W_g(t,f)+|b^2|*W_s(t,f)+\int_{-\infty}^\infty [ab^*g(t+\tau/2)s^*(t-\tau/2)+a^*bg^*(t-\tau/2)s(t+\tau/2)]e^{-j2\pi\tau\,f} \, d\tau$


## 參考

• Jian-Jiun Ding, Time frequency analysis and wavelet transform lecture note, National Taiwan University (NTU), Taipei, Taiwan, 2012.
• S. C. Pei and J. J. Ding, “Relations between Gabor transforms and fractional Fourier transforms and their applications for signal processing,” IEEE Trans.Signal Processing, vol. 55, no. 10, pp. 4839-4850, Oct. 2007.