**模糊函數**(**A**mbiguity **f**unction,**AF**):  

**解析失败 (带SVG或PNG备选的MathML（建议用于现代的浏览器和辅助工具）：从服务器"/mathoid/local/v1/"返回无效的响应（"Math extension cannot connect to Restbase."）：): {\displaystyle AF\_s(\theta,\tau)=\int\_{-\infty}^{\infty} s\left(t+\frac{\tau}{2}\right)s^\*\left(t-\frac{\tau}{2}\right)e^{-j\theta t}\, dt}**

**韋格納分佈**(**W**igner **d**istribution,**WD**):  

{\displaystyle WD\_{s}(t,\omega )=\int \_{-\infty }^{\infty }s\left(t+{\frac {\tau }{2}}\right)s^{\*}\left(t-{\frac {\tau }{2}}\right)e^{-j\omega \tau }\,d\tau }

模糊函數與韋格納分佈關係
------------

一個訊號s(t)，自相關函數為
{\displaystyle R(\tau )=\int \_{-\infty }^{\infty }s(t)s^{\*}(t-\tau )\,dt}
如果{\displaystyle R(\tau )}為時間相依性(time-dependent)，則時間相依自相關(time-dependent auto-correlation)為{\displaystyle R(t,\tau )}，
時間相依(時變)頻譜(time-dependent spectrum)可以表示的形式類似於傳統的功率譜，即對時間相依自相關函數做傅立葉變換。  

{\displaystyle P(t,\omega )=\int \_{-\infty }^{\infty }R(t,\tau )e^{-j\omega \tau }\,d\tau }   

不同的時間相依自相關會導致不同的時間相依功率譜。  

如果
{\displaystyle R(t,\tau )=s\left(t+{\frac {\tau }{2}}\right)s^{\*}\left(t-{\frac {\tau }{2}}\right)} 
，則時間相依功率譜變成為Wigner distribution  

若對{\displaystyle R(t,\tau )}中的t做傅立葉逆轉換，得到另一個時頻表示，**對稱模糊函數(symmetric ambiguity function,SAF)**  

{\displaystyle SAF\_{s}(\theta ,\tau )={\frac {1}{2\pi }}\int \_{-\infty }^{\infty }s\left(t+{\frac {\tau }{2}}\right)s^{\*}\left(t-{\frac {\tau }{2}}\right)e^{j\theta t}\,dt}
模糊函數反映信號在時間和相位的相關性，並已廣泛應用在雷達和聲納系統上。
給一個對稱模糊函數{\displaystyle SAF\_{s}(\theta ,\tau )}，透過傅立葉變換可以得到時間相依自相關:  

{\displaystyle \int \_{-\infty }^{\infty }SAF\_{s}(\theta ,\tau )e^{-j\theta t}\,d\theta =s\left(t+{\frac {\tau }{2}}\right)s^{\*}\left(t-{\frac {\tau }{2}}\right)}
由上式可以推得  

{\displaystyle WD\_{s}(t,\omega )=\int \_{-\infty }^{\infty }\int \_{-\infty }^{\infty }SAF\_{s}(\theta ,\tau )e^{-j(\omega \tau +\theta t)}\,d\theta \,d\tau }  

也就是對**對稱模糊函數**做兩次傅立葉變換可以得到Wigner distribution

範例
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一個訊號為兩個Gaussian函數的和:  

{\displaystyle s(t)=\sum \_{i=1}^{2}s\_{i}(t)=\sum \_{i=1}^{2}{\sqrt[{4}]{\frac {\alpha }{\pi }}}e^{-{\tfrac {\alpha }{2}}(t-t\_{i})^{2}+j\omega \_{i}t}}  

{\displaystyle \Rightarrow SAF\_{s}(\theta ,\tau )=\sum \_{i=1}^{2}SAF\_{si}(\theta ,\tau )+SAF\_{s1,s2}(\theta ,\tau )+SAF\_{s2,s1}(\theta ,\tau )}  

* 其中{\displaystyle SAF\_{s1}(\theta ,\tau ),SAF\_{s2}(\theta ,\tau )}集中在原點(0,0)，而{\displaystyle SAF\_{s1,s2}(\theta ,\tau )}集中在{\displaystyle (t\_{1}-t\_{2},\omega \_{1}-\omega \_{2})}，而{\displaystyle SAF\_{s2,s1}(\theta ,\tau )}相似於{\displaystyle SAF\_{s1,s2}(\theta ,\tau )}，除了中心點在{\displaystyle (t\_{2}-t\_{1},\omega \_{2}-\omega \_{1})}
	+ {\displaystyle SAF\_{s1,s2}(\theta ,\tau )=e^{-{\tfrac {1}{4\alpha }}(\theta -\omega \_{d})^{2}+{\tfrac {\alpha }{4}}(\tau -t\_{d})^{2}}e^{j(\omega \_{u}\tau -\theta t\_{u}+\omega \_{d}t\_{u})}} , {\displaystyle t\_{u}={\tfrac {t\_{1}+t\_{2}}{2}}} , {\displaystyle \omega \_{u}={\tfrac {\omega \_{1}+\omega \_{2}}{2}}} , {\displaystyle t\_{d}=t\_{1}-t\_{2}} , {\displaystyle \omega \_{d}=\omega \_{1}-\omega \_{2}}

### 模糊域(ambiguity domain)的auto-term與cross-term

從範例中得知一項重要事實，即為，在模糊域(ambiguity domain)中的auto-term總是集中在原點(0,0)，而cross-term總是在遠離原點處，所以可以用一個2D lowpass filter在模糊域中抑制cross-term的干擾，如下:  

{\displaystyle \int \_{-\infty }^{\infty }\int \_{-\infty }^{\infty }SAF\_{s}(\theta ,\tau )\Phi (\theta ,\tau )e^{-j(\theta t+\omega \tau )}\,d\theta \,d\tau } ,其中{\displaystyle \Phi (\theta ,\tau )}為2D lowpass filter  

### 兩高斯信號和之模糊函數與韋格納分佈應對關係

如果{\displaystyle \int \_{-\infty }^{\infty }\int \_{-\infty }^{\infty }\Phi (\theta ,\tau )\,d\theta \,d\tau =\phi (t,\omega )}，則  

{\displaystyle \int \_{-\infty }^{\infty }\int \_{-\infty }^{\infty }SAF\_{s}(\theta ,\tau )\Phi (\theta ,\tau )e^{-j(\theta t+\omega \tau )}\,d\theta \,d\tau }  

{\displaystyle =\int \_{-\infty }^{\infty }\int \_{-\infty }^{\infty }\phi (x,y)WD\_{s}(t-x,\omega -y)\,dx\,dy=SWD(t,\omega )}  

* 其中SWD為smoothed Wigner distribution

通常{\displaystyle \Phi (\theta ,\tau )}( 和{\displaystyle \phi (t,\omega )} )當作kernal function，用來控制SWD的特性。  

  

若Wigner分佈和對稱模糊函數用大小(magnitude)及相位(phase)表示，如下:  

{\displaystyle WD\_{s1,s2}(t,\omega )=A\_{WD}(t,\omega )e^{j\varphi \_{WD}(t,\omega )}}   

{\displaystyle SAF\_{s1,s2}(\theta ,\tau )=A\_{SAF}(\theta ,\tau )e^{j\varphi \_{SAF}(\theta ,\tau )}}   

而{\displaystyle {\frac {\partial }{\partial \theta }}\varphi \_{SAF}(\theta ,\tau )=-t\_{u}} , {\displaystyle {\frac {\partial }{\partial \tau }}\varphi \_{SAF}(\theta ,\tau )=\omega \_{u}}  

也就是說對**對稱模糊函數**的相位做偏微分，會等於Wigner分佈的時頻(time-frequency)中心。  

相反地，{\displaystyle {\frac {\partial }{\partial \omega }}\varphi \_{WD}(t,\omega )=t\_{d}} , {\displaystyle {\frac {\partial }{\partial t}}\varphi \_{WD}(t,\omega )=\omega \_{d}}  

則為對Wigner分佈的相位做偏微分，會等於對稱模糊函數的中心。  

  

如果{\displaystyle \omega \_{1}=\omega \_{2}=\omega \_{0}}，則  

{\displaystyle SAF\_{s1,s2}(\theta ,\tau )=e^{-[{\tfrac {1}{4\alpha }}\theta ^{2}+{\tfrac {\alpha }{4}}(\tau -t\_{d})^{2}]}e^{j(\omega \_{0}\tau -\theta t\_{u})}}   

會集中在\tau 軸上。  

  

如果{\displaystyle t\_{1}=t\_{2}=t\_{0}}，則  

{\displaystyle SAF\_{s1,s2}(\theta ,\tau )=e^{-[{\tfrac {1}{4\alpha }}(\theta -\omega \_{d})^{2}+{\tfrac {\alpha }{4}}\tau ^{2}]}e^{j[\omega \_{u}\tau -(\theta -\omega \_{d})t\_{0}]}}   

會集中在\theta 軸上。

參考
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* Weiss, Lora G. "Wavelets and Wideband Correlation Processing". IEEE Signal Processing Magazine, pp. 13–32, Jan 1994
* Shie Qian, *Introduction to time-frequency and wavelet transforms*, Upper Saddle River, NJ : Prentice Hall, c2002
* L. Sibul, L. Ziomek, "Generalised wideband crossambiguity functiom", IEEE International Conference on Acoustics, Speech, and Signal Processing, ICASSP '81.01/05/198105/1981; 6:1239- 1242.