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. ${\displaystyle D_{x}(t,f)=G_{x}(t,f)\times W_{x}(t,f)}$

1.  :${\displaystyle G_{x}(t,f)=\int _{-\infty }^{\infty }e^{-\pi (\tau -t)^{2}}e^{-j2\pi f\tau }x(\tau )\,d\tau }$
2.  :${\displaystyle W_{x}(t,f)=\int _{-\infty }^{\infty }x(t+\tau /2)x^{*}(t-\tau /2)e^{-j2\pi \tau \,f}\,d\tau }$

1. ${\displaystyle D_{x}(t,f)=\min \left\{|G_{x}(t,f)|^{2},|W_{x}(t,f)|\right\}}$
2. ${\displaystyle D_{x}(t,f)=W_{x}(t,f)\times \{|G_{x}(t,f)|>0.25\}}$
3. ${\displaystyle D_{x}(t,f)=G_{x}^{2.6}(t,f)W_{x}^{0.7}(t,f)}$

特性

不會有cross-term問題

cross-term問題主要發生在韋格納轉換(${\displaystyle W_{x}(t,f)}$)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${\displaystyle x(t)=ag(t)+bs(t)}$ 根據定義:

    ${\displaystyle W_{x}(t,f)=\int _{-\infty }^{\infty }x(t+\tau /2)x^{*}(t-\tau /2)e^{-j2\pi \tau \,f}\,d\tau }$


    ${\displaystyle W_{x}(t,f)=\int _{-\infty }^{\infty }[ag(t+\tau /2)+bs(t+\tau /2)][a^{*}g(t-\tau /2)+b^{*}s(t+\tau /2)]e^{-j2\pi \tau \,f}\,d\tau }$
${\displaystyle W_{x}(t,f)=|a^{2}|*W_{g}(t,f)+|b^{2}|*W_{s}(t,f)+\int _{-\infty }^{\infty }[ab^{*}g(t+\tau /2)s^{*}(t-\tau /2)+a^{*}bg^{*}(t-\tau /2)s(t+\tau /2)]e^{-j2\pi \tau \,f}\,d\tau }$


加伯–韋格納轉換實現方法(簡化技巧)

(1) 當${\displaystyle G_{X}(t,f)\approx 0,D_{x}(t,f)=G_{x}^{\alpha }(t,f)W_{x}^{\beta }(t,f)\approx 0}$

${\displaystyle W_{x}(t,f)}$ 只需算${\displaystyle G_{x}(t,f)}$ 不近似於0的地方

(2)當 x(t) 是實函數時，對加伯轉換而言

${\displaystyle X(f)=X^{*}(-f)}$

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.
• Jian-Jiun Ding, Time frequency analysis and wavelet transform class notes, Graduate Institute of Communication Engineering, National Taiwan University (NTU), Taipei, Taiwan, 2017.