S轉換

S轉換(s-transform)，或S變換是一種時頻分析的工具。

定義

${\displaystyle S_{x}(t,\,f)=\int _{-\infty }^{\infty }x(\tau )\,|f|\,e^{-\pi (t-\tau )^{2}f^{2}}e^{-j2\pi f\tau }\,d\tau }$

${\displaystyle w(t,f)=|f|e^{-\pi t^{2}f^{2}}}$

另種表示-頻譜表示式

${\displaystyle x(t)\ast h(t)={\mathcal {F}}^{-1}(X(f)\cdot H(f))}$

S轉換能以頻域 ${\displaystyle X(f)}$ 表示，

${\displaystyle S_{x}(t,f)=\int _{-\infty }^{\infty }(x(\tau )e^{-j2\pi f\tau })(|f|e^{-\pi (t-\tau )^{2}f^{2}})\,d\tau }$

${\displaystyle x(\tau )e^{-j2\pi f\tau }}$ 以及 ${\displaystyle |f|e^{-\pi (t-\tau )^{2}f^{2}}}$分別取傅立葉變換可得

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

逆S轉換(inverse S-transform)

S轉換可以沿著時間軸方向積分，將可以得到${\displaystyle x(t)}$的頻譜${\displaystyle X(f)}$。推導如下，

${\displaystyle \int _{-\infty }^{\infty }|f|e^{-\pi (t-\tau )^{2}f^{2}}\,dt=|f|\int _{-\infty }^{\infty }e^{-\pi (t-\tau )^{2}f^{2}}\,dt=1}$

${\displaystyle \int _{-\infty }^{\infty }S_{x}(t,f)\,dt=\int _{-\infty }^{\infty }x(\tau )\left[\int _{-\infty }^{\infty }|f|\,e^{-\pi (t-\tau )^{2}f^{2}}\,dt\right]\,e^{-j2\pi f\tau }\,d\tau =X(f)}$

${\displaystyle x(\tau )=\int _{-\infty }^{\infty }\left[\int _{-\infty }^{\infty }S_{x}(t,f)\,dt\right]\,e^{j2\pi f\tau }\,df}$
${\displaystyle =\int _{-\infty }^{\infty }X(f)\,e^{j2\pi f\tau }\,df}$

濾波應用(Filtering)

S轉換如同其他時頻分析轉換，皆可以設計波器來達到消除雜訊留下訊號的功用，

${\displaystyle x_{filter}(\tau )=\int _{-\infty }^{\infty }\left[\int _{-\infty }^{\infty }S_{x}(t,f)\cdot U(t,f)\,dt\right]\,e^{j2\pi f\tau }\,df}$

離散時間S轉換

S轉換相較於加伯轉換，雖在清晰度有較好的改善，但也有其缺點，就是運算複雜度變高，積分的範圍會隨著${\displaystyle f\,}$的增加而增加。

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

${\displaystyle t=n\Delta _{T}\,\,f=m\Delta _{F}\,\,\alpha =p\Delta _{F}}$

${\displaystyle \Delta _{T}\,}$表示取樣時間間隔${\displaystyle ,\Delta _{F}\,}$表示取樣頻率

${\displaystyle \Delta _{T}\cdot \Delta _{F}=1/N}$

${\displaystyle X[m\Delta _{F}]={\frac {1}{N}}\,\sum _{k=0}^{N-1}x[k\Delta _{T}]\,e^{\frac {-j2\pi mk}{N}}}$

${\displaystyle S_{x}(n\Delta _{T}\,,m\Delta _{F})=\sum _{p=0}^{N-1}X[(p+m)\,\Delta _{F}]\,e^{-\pi {\frac {p^{2}}{m^{2}}}}\,e^{\frac {j2pn}{N}}}$

${\displaystyle S_{x}(n\Delta _{T}\,,0)={\frac {1}{N}}\,\sum _{k=0}^{N-1}x[k\Delta _{F}}$]

流程

Step1 : 計算${\displaystyle X[p\Delta _{F}]\,}$,這個步驟只需要計算一次。
Step2 : 計算${\displaystyle e^{-\pi {\frac {p^{2}}{m^{2}}}}}$for ${\displaystyle f=m\Delta _{F}}$
Step3 : 將${\displaystyle X[p\Delta _{F}]}$移動至${\displaystyle X[(p+m)\Delta _{F}]}$
Step4 : 將Step2,Step3的結果相乘得到

${\displaystyle B[m,p]=X[(p+m)\Delta _{F}]\cdot e^{-\pi {\frac {p^{2}}{m^{2}}}}}$

Step5 : 對B[m,p]取逆離散傅立葉變換(IDFT)可得到，${\displaystyle S_{x}(n\Delta _{T}\,,m\Delta _{F})}$${\displaystyle f=m\Delta _{F}}$的行向量
Step6 : 重複Step2~5直到${\displaystyle S_{x}(n\Delta _{T}\,,m\Delta _{F})}$全部定義完成。

S轉換特性

S轉換與加伯轉換(Gabor Transform)很相似，

${\displaystyle G_{x}(t,f)=\int _{-\infty }^{\infty }x(\tau )\,e^{-\pi (t-\tau )^{2}}e^{-j2\pi f\tau }\,d\tau }$
${\displaystyle S_{x}(t,f)=\int _{-\infty }^{\infty }x(\tau )\,|f|e^{-\pi (t-\tau )^{2}f^{2}}e^{-j2\pi f\tau }\,d\tau }$

 低頻 時域解析度差 頻域解析度佳 高頻 頻域解析度差 時域解析度佳

S轉換一般式

${\displaystyle S_{x}(t,f)=|s(f)|\int _{-\infty }^{\infty }x(\tau )\,e^{-\pi (t-\tau )^{2}s^{2}(f)}e^{-j2\pi f\tau }\,d\tau }$
${\displaystyle s(f)}$是一個相對平緩的曲線(見底下示意圖)，當${\displaystyle f\rightarrow 0}$時，${\displaystyle s(f)\neq 0}$

S轉換是一種運算量高的時頻分析工具，尤其在低頻部分，Gaussian Window寬度變寬，頻域解析度比加伯轉換來的好，所以S轉換對於低頻訊號分析比較有優勢

與韋格納分佈的比較

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

與加伯轉換的比較

${\displaystyle G_{x}(t,f)=\int _{-\infty }^{\infty }x(\tau )e^{\left[-\pi (t-\tau )^{2}f^{2}\right]}e^{\left(-j2\pi f\tau \right)}d\tau }$

與小波轉換的關係

${\displaystyle W(\tau ,\,d)=\int _{-\infty }^{\infty }x(t)\,W(t-\tau ,\,d))\,dt}$

${\displaystyle S(\tau ,\,f)=e^{-j2\pi \tau f}W(\tau ,\,d)}$

${\displaystyle w(t,\,f)=|f|\,\exp[-\pi t^{2}f^{2}]\exp[-j2\pi ft]}$

S轉換，加伯轉換和短時距傅立葉轉換的比較

${\displaystyle x(t)=cos(2\pi t),t<10,}$

${\displaystyle x(t)=cos(2\pi t),10\leq t<20,}$

${\displaystyle x(t)=cos(2\pi t),20\leq t.}$

參考文獻

• R. G. Stockwell, L. Mansinha, and R. P. Lowe, "Localization of the complex spectrum: the S transform," IEEE Trans. Signal Processing, vol. 44, no. 4, pp. 998–1001, Apr. 1996.
• Jian-Jiun Ding, Time frequency analysis and wavelet transform class note, the Department of Electrical Engineering, National Taiwan University (NTU), Taipei, Taiwan, 2010.
• Sitanshu Sekhar, Ganapati Panda and Nithin V George, "An Improved S-Transform for Time-Frequency Analysis," "IACC2009", pp. 315-319, March 2009.