重新分布法

時頻譜

${\displaystyle SP_{x}(t,f)=|X(t,f)|^{2}=\left|\int _{-\infty }^{\infty }w(t-\tau )x(\tau )e^{-j2\pi f\tau }d\tau \right|^{2}.}$

方法

${\displaystyle h_{\omega }(t)=h(t)e^{j\omega t}}$

{\displaystyle {\begin{aligned}\epsilon (t,\omega )&=\int x(\tau )h(t-\tau )e^{-j\omega \left[\tau -t\right]}d\tau \\&=e^{j\omega t}\int x(\tau )h(t-\tau )e^{-j\omega \tau }d\tau \\&=e^{j\omega t}X(t,\omega )\\&=X_{t}(\omega )\\&=M_{t}(\omega )e^{j\phi _{\tau }(\omega )}.\end{aligned}}}

${\displaystyle x(t)}$可透過滑動視窗的係數以下式重建：

{\displaystyle {\begin{aligned}x(t)&=\iint X_{\tau }(\omega )h_{\omega }^{*}(\tau -t)d\omega d\tau \\&=\iint X_{\tau }(\omega )h(\tau -t)e^{-j\omega \left[\tau -t\right]}d\omega d\tau \\&=\iint M_{\tau }(\omega )e^{j\phi _{\tau }(\omega )}h(\tau -t)e^{-j\omega \left[\tau -t\right]}d\omega d\tau \\&=\iint M_{\tau }(\omega )h(\tau -t)e^{j\left[\phi _{\tau }(\omega )-\omega \tau +\omega t\right]}d\omega d\tau .\end{aligned}}}

{\displaystyle {\begin{aligned}{\frac {\partial }{\partial \omega }}\left[\phi _{\tau }(\omega )-\omega \tau +\omega t\right]&=0,\\{\frac {\partial }{\partial \tau }}\left[\phi _{\tau }(\omega )-\omega \tau +\omega t\right]&=0.\end{aligned}}}

{\displaystyle {\begin{aligned}{\hat {t}}(\tau ,\omega )&=\tau -{\frac {\partial \phi _{\tau }(\omega )}{\partial \omega }}=-{\frac {\partial \phi (\tau ,\omega )}{\partial \omega }},\\{\hat {\omega }}(\tau ,\omega )&={\frac {\partial \phi _{\tau }(\omega )}{\partial \tau }}=\omega +{\frac {\partial \phi (\tau ,\omega )}{\partial \tau }}.\end{aligned}}}

快速時頻重新分布法

{\displaystyle {\begin{aligned}{\frac {\partial \phi (t,\omega )}{\partial t}}&\approx {\frac {1}{\Delta t}}\left[\phi \left(t+{\frac {\Delta t}{2}},\omega \right)-\phi \left(t-{\frac {\Delta t}{2}},\omega \right)\right]\\{\frac {\partial \phi (t,\omega )}{\partial \omega }}&\approx {\frac {1}{\Delta \omega }}\left[\phi \left(t,\omega +{\frac {\Delta \omega }{2}}\right)-\phi \left(t,\omega -{\frac {\Delta \omega }{2}}\right)\right]\end{aligned}}}

{\displaystyle {\begin{aligned}{\hat {t}}(t,\omega )&=t-{\frac {\iint \tau \cdot W_{x}(t-\tau ,\omega -\nu )\cdot \Phi (\tau ,\nu )d\tau d\nu }{\iint W_{x}\left(t-\tau ,\omega -\nu \right)\cdot \Phi (\tau ,\nu )d\tau d\nu }}\\{\hat {\omega }}(t,\omega )&=\omega -{\frac {\iint \nu \cdot W_{x}(t-\tau ,\omega -\nu )\cdot \Phi (\tau ,\nu )d\tau d\nu }{\iint W_{x}(t-\tau ,\omega -\nu )\cdot \Phi (\tau ,\nu )d\tau d\nu }}\end{aligned}}}

{\displaystyle {\begin{aligned}{\hat {t}}(t,\omega )&=t-\Re \left\{{\frac {X_{{\mathcal {T}}h}(t,\omega )\cdot X^{*}(t,\omega )}{|X(t,\omega )|^{2}}}\right\}\\{\hat {\omega }}(t,\omega )&=\omega +\Im \left\{{\frac {X_{{\mathcal {D}}h}(t,\omega )\cdot X^{*}(t,\omega )}{|X(t,\omega )|^{2}}}\right\}\end{aligned}}}

可分離性

「open」一字之語音訊號透過較長之分析視窗重新分布後得到的时频谱。其結果為利用長度為54.4毫秒、外型參數${\displaystyle \alpha }$為9的凯泽窗計算所得，圖中不同諧波間的分離較為明顯。
「open」一字之語音訊號透過較短之分析視窗重新分布後得到的时频谱。其結果為利用長度為13.6毫秒、外型參數${\displaystyle \alpha }$為9的凯泽窗計算所得，圖中不同聲門脈衝間的分離較為明顯。

${\displaystyle x(t)=\sum _{n}A_{n}(t)e^{j\theta _{n}(t)}}$

${\displaystyle \omega _{n}(t)={\frac {d\theta _{n}(t)}{dt}}}$

{\displaystyle {\begin{aligned}\omega _{n}(t)&={\frac {\partial }{\partial t}}\arg\{x_{n}(t)\}\\&={\frac {\partial }{\partial t}}\arg\{X(t,\omega _{0})\}\end{aligned}}}

參考資料

1. ^ Hainsworth, Stephen. Chapter 3: Reassignment methods. Techniques for the Automated Analysis of Musical Audio (PhD). University of Cambridge. 2003. （原始内容存档于2013-05-24）.
2. F. Auger & P. Flandrin. Improving the readability of time-frequency and time-scale representations by the reassignment method. IEEE Transactions on Signal Processing. May 1995, 43 (5): 1068–1089. doi:10.1109/78.382394.
3. ^ P. Flandrin, F. Auger, and E. Chassande-Mottin, Time-frequency reassignment: From principles to algorithms, in Applications in Time-Frequency Signal Processing (A. Papandreou-Suppappola, ed.), ch. 5, pp. 179 – 203, CRC Press, 2003.
4. ^ K. Kodera; R. Gendrin & C. de Villedary. Analysis of time-varying signals with small BT values. IEEE Transactions on Acoustics, Speech and Signal Processing. Feb 1978, 26 (1): 64–76. doi:10.1109/TASSP.1978.1163047.
5. D. J. Nelson. Cross-spectral methods for processing speech. Journal of the Acoustical Society of America. Nov 2001, 110 (5): 2575–2592. doi:10.1121/1.1402616.

延伸閱讀

• S. A. Fulop and K. Fitz, A spectrogram for the twenty-first century, Acoustics Today, vol. 2, no. 3, pp. 26–33, 2006.
• S. A. Fulop and K. Fitz, Algorithms for computing the time-corrected instantaneous frequency (reassigned) spectrogram, with applications, Journal of the Acoustical Society of America, vol. 119, pp. 360 – 371, Jan 2006.