**科恩系列分佈**（Cohen's class distribution）於1966年由L. Cohen首次提出，且其使用雙線性轉換亦是此種轉換形式中最通用的一種。在幾種常見的時頻分佈中，Cohen's class分佈是最強大的轉換之一。隨著近幾年來時頻分析發展，應用也越來越多元。Cohen's class分佈和短時距傅立葉變換比較起來有較高的清晰度，但也相對的有交叉項(cross-term)的問題，不過可選擇適當的遮罩函數(mask function)來將交叉項的問題降到最低。

數學定義
----

{\displaystyle C\_{x}(t,f)=\int \_{-\infty }^{\infty }\int \_{-\infty }^{\infty }A\_{x}(\eta ,\tau )\Phi (\eta ,\tau )\exp(j2\pi (\eta t-\tau f))\,d\eta \,d\tau ,} ，
其中 {\displaystyle A\_{x}\left(\eta ,\tau \right)=\int \_{-\infty }^{\infty }x(t+\tau /2)x^{\*}(t-\tau /2)e^{-j2\pi t\eta }\,dt.} 為模糊函數(Ambiguity Function) ，且\Phi \left(\eta ,\tau \right)為一遮罩函數，通常是低通函數用來濾除雜訊。

科恩系列分佈函數
--------

### 韋格納分布(Wigner Distribution Function)

當Cohen's class分佈中的{\displaystyle \Phi \left(\eta ,\tau \right)=1}時，Cohen's class分佈會成韋格納分布(Wigner distribution function){\displaystyle W\_{x}(t,f)=\int \_{-\infty }^{\infty }x(t+\tau /2)x^{\*}(t-\tau /2)e^{-j2\pi f\tau }\,d\tau }。
利用韋格納分佈對函數{\displaystyle x(t)=exp(j0.015t^{4}+j0.06t^{3}-j0.3t^{2}+jt)}作時頻分析的結果可見右圖。
### 錐狀分布(Cone-Shape Distribution)

當Cohen's class分佈中的{\displaystyle \phi (t,\tau )={\frac {1}{\left|\tau \right|}}exp(-2\pi \alpha \tau ^{2})\Pi ({\frac {t}{\tau }})}，且{\displaystyle \Phi \left(\eta ,\tau \right)=sinc(\eta \tau )exp((-2\pi \alpha \tau ^{2})}時，
其中{\displaystyle \phi (t,\tau )=\int \_{-\infty }^{\infty }\Phi (\eta ,\tau )exp(j2\pi \eta t)d\eta }，Cohen's class分佈會成錐狀分布。
右圖為不同的\alpha 值下的錐狀分佈時頻分析圖。

  

### 喬伊-威廉斯(Choi-Williams)

當Cohen's class分佈中的{\displaystyle \Phi \left(\eta ,\tau \right)=exp[-\alpha (\eta \tau )^{2}]}時，Cohen's class分佈會成喬伊-威廉斯分布。
右圖為不同的\alpha 值下的錐狀分佈時頻分析圖。

科恩系列分佈優缺點
---------

優點:
1.可選擇適當的遮罩函數來避免掉交叉項問題 。
2.具有高清晰度。
缺點
1. 需要較高的計算量與時間。
2. 缺乏良好的數學特性。

科恩系列分佈的實現
---------

{\displaystyle C\_{x}(t,f)=\int \_{-\infty }^{\infty }\int \_{-\infty }^{\infty }A\_{x}(\eta ,\tau )\Phi (\eta ,\tau )\exp(j2\pi (\eta t-\tau f))\,d\eta \,d\tau ,}
{\displaystyle =\int \_{-\infty }^{\infty }\int \_{-\infty }^{\infty }\int \_{-\infty }^{\infty }x(u+{\frac {\tau }{2}})x^{\*}(u-{\frac {\tau }{2}})\Phi (\eta ,\tau )e^{-j2\pi u\eta +j2\pi (\eta t-\tau f)}dud\tau d\eta }
### 簡化方法一:不是所有的{\displaystyle A\_{x}(\eta ,\tau )}的值都要計算出

對{\displaystyle \ \left|\eta \right|>B\ }或{\displaystyle \ \left|\tau \right|>C}，若{\displaystyle \Phi (\eta ,\tau )=0}，則{\displaystyle C\_{x}(t,f)=\int \_{-C}^{C}\int \_{-B}^{B}\int \_{-\infty }^{\infty }x(u+{\frac {\tau }{2}})x^{\*}(u-{\frac {\tau }{2}})\Phi (\eta ,\tau )e^{-j2\pi u\eta +j2\pi (\eta t-\tau f)}dud\tau d\eta }
### 簡化方法二:注意，\eta 這個參數和輸入及輸出都無關

{\displaystyle C\_{x}(t,f)=\int \_{-C}^{C}\int \_{-\infty }^{\infty }x(u+{\frac {\tau }{2}})x^{\*}(u-{\frac {\tau }{2}})[\int \_{-B}^{B}\Phi (\eta ,\tau )e^{-j2\pi ,\eta (t-u)}d\eta ]e^{-j2\pi \tau ,f}dud\tau }
{\displaystyle =\int \_{-C}^{C}\int \_{-\infty }^{\infty }x(u+{\frac {\tau }{2}})x^{\*}(u-{\frac {\tau }{2}})\Phi (\tau ,t-u)e^{-j2\pi \tau ,f}dud\tau }，其中
{\displaystyle \Phi (\tau ,t-u)=\int \_{-B}^{B}\Phi (\eta ,\tau )e^{-j2\pi ,\eta (t-u)}d\eta }，由於{\displaystyle \Phi (\tau ,t-u)}和輸入無關，可事先算出，因此可簡化成兩個積分式。
### 簡化方法三:使用摺積方法(convolution)

{\displaystyle C\_{x}(t,f)=\int \_{-\infty }^{\infty }\int \_{-\infty }^{\infty }x(u+{\frac {\tau }{2}})x^{\*}(u-{\frac {\tau }{2}})\phi (t-u,\tau )due^{-j2\pi f\tau }d\tau }，其中
{\displaystyle \phi (t,\tau )=\int \_{-\infty }^{\infty }\Phi (\eta ,\tau )exp(j2\pi \eta t)d\eta }。對{\displaystyle \left|t\right|>b}或是{\displaystyle \left|\tau \right|>c}，則
{\displaystyle C\_{x}(t,f)=\int \_{-c}^{c}\int \_{t-b}^{t+b}x(u+{\frac {\tau }{2}})x^{\*}(u-{\frac {\tau }{2}})\phi (t-u,\tau )due^{-j2\pi f\tau }d\tau }，上式為一摺積式。

模糊函數 (Ambiguity Function)
-------------------------

模糊函數的定義為：

{\displaystyle A\_{x}\left(\eta ,\tau \right)=\int \_{-\infty }^{\infty }x(t+{\tfrac {\tau }{2}})x^{\*}(t-{\tfrac {\tau }{2}})e^{-j2\pi t\eta }\,dt}
### Modulation 和 Time Shifting 對模糊函數的影響

我們來看一下 x(t) 對於模糊函數的影響

(1) 假設 x\_{1}(t) 是一個高斯函數: {\displaystyle ae^{-(t-b)^{2}/2c^{2}}}, 其中 {\displaystyle a=1,b=0,c={\sqrt {\tfrac {1}{2\alpha }}}}

那麼我們可以得到 {\displaystyle x\_{1}(t)=e^{-\alpha \pi t^{2}}}, 代入模糊函數 {\displaystyle A\_{x}\left(\eta ,\tau \right)} 中：

{\displaystyle A\_{x\_{1}}\left(\eta ,\tau \right)=\textstyle \int \limits \_{-\infty }^{\infty }\displaystyle e^{-\alpha \pi {(t+{\tfrac {\tau }{2}})}^{2}}\ e^{-\alpha \pi {(t-{\tfrac {\tau }{2}})}^{2}}\ e^{-j2\pi t\eta }\ dt}
{\displaystyle =\textstyle \int \limits \_{-\infty }^{\infty }\displaystyle e^{-\alpha \pi (2t^{2}+{\tfrac {\tau }{2}}^{2})}\ e^{-j2\pi t\eta }\ dt}
(2) 假設 x\_{2}(t) 是一個經過 shifting 和 modulation 的高斯函數: 

那麼我們可以得到 {\displaystyle x\_{2}(t)=e^{-\alpha \pi (t-t\_{0})^{2}+j2\pi f\_{0}t}}, 代入模糊函數 {\displaystyle A\_{x}\left(\eta ,\tau \right)} 中：

{\textstyle A\_{x\_{2}}\left(\eta ,\tau \right)=\textstyle \int \limits \_{-\infty }^{\infty }\displaystyle e^{-\alpha \pi {(t+{\tfrac {\tau }{2}}-t\_{0})}^{2}+j2\pi f\_{0}(t+{\tfrac {\tau }{2}})}\ e^{-\alpha \pi {(t-{\tfrac {\tau }{2}}-t\_{0})}^{2}-j2\pi f\_{0}(t-{\tfrac {\tau }{2}})}\ e^{-j2\pi t\eta }\ dt}
{\displaystyle =\textstyle \int \limits \_{-\infty }^{\infty }\displaystyle e^{-\alpha \pi (2(t-t\_{0})^{2}+{\tau /2}^{2})+j2\pi f\_{0}\tau }\ e^{-j2\pi t\eta }\ dt}
{\displaystyle =\textstyle \int \limits \_{-\infty }^{\infty }\displaystyle e^{-\alpha \pi (2t^{2}+{\tfrac {\tau ^{2}}{2}})}\ e^{j2\pi f\_{0}\tau }\ e^{-j2\pi t\eta }\ e^{-j2\pi t\_{0}\eta }\ dt}
我們可以看到 {\displaystyle |A\_{x\_{1}}\left(\tau ,\eta \right)|=|A\_{x\_{2}}\left(\tau ,\eta \right)|}, 

因此我們可以得出 time shifting t\_{0} 和 modulation f\_0 並不會影響 {\displaystyle |A\_{x}\left(\tau ,\eta \right)|}

積分後，{\displaystyle A\_{x}\left(\tau ,\eta \right)={\sqrt {\tfrac {1}{2\alpha }}}e^{-\pi ({\tfrac {\alpha \tau ^{2}}{2}}+{\tfrac {\eta ^{2}}{2\alpha }})}e^{j2\pi (f\_{0}\tau -t\_{0}\eta )}}

所以 {\displaystyle A\_{x}\left(\tau ,\eta \right)} 在 {\displaystyle \tau =0,\eta =0} 的地方會有最大的 {\displaystyle |A\_{x}\left(\tau ,\eta \right)|}

### 交叉項 Cross-term 問題

上述所列出來的是當 x(t) 只有一項而已 (one term only)，如果 x(t) 有兩項以上的元素構成 (more than two terms), {\displaystyle x(t)=x\_{1}(t)+x\_{2}(t)+\cdot \cdot \cdot +x\_{n}(t)}，依然會有交叉項 (cross-term) 的問題存在。

假設 {\displaystyle x(t)=x\_{1}(t)+x\_{2}(t)} , 其中

{\displaystyle {\begin{cases}x\_{1}(t)=e^{-\alpha \_{1}\pi (t-t\_{1})^{2}+j2\pi f\_{1}t}\\x\_{2}(t)=e^{-\alpha \_{2}\pi (t-t\_{2})^{2}+j2\pi f\_{2}t}\end{cases}}}
將 {\displaystyle x(t)} 代入模糊函數 {\displaystyle A\_{x}\left(\eta ,\tau \right)=\int \_{-\infty }^{\infty }x(t+{\tfrac {\tau }{2}})x^{\*}(t-{\tfrac {\tau }{2}})e^{-j2\pi t\eta }\,dt} 中：

{\displaystyle A\_{x}\left(\eta ,\tau \right)=\underbrace {\int \_{-\infty }^{\infty }x\_{1}(t+{\tfrac {\tau }{2}})x\_{1}^{\*}(t-{\tfrac {\tau }{2}})e^{-j2\pi t\eta }\,dt} \_{A\_{x\_{1}}(\tau ,\eta )}+\underbrace {\int \_{-\infty }^{\infty }x\_{2}(t+{\tfrac {\tau }{2}})x\_{2}^{\*}(t-{\tfrac {\tau }{2}})e^{-j2\pi t\eta }\,dt} \_{A\_{x\_{2}}(\tau ,\eta )}}
{\displaystyle +\underbrace {\int \_{-\infty }^{\infty }x\_{1}(t+{\tfrac {\tau }{2}})x\_{2}^{\*}(t-{\tfrac {\tau }{2}})e^{-j2\pi t\eta }\,dt} \_{A\_{{x\_{1}}{x\_{2}}}(\tau ,\eta )}+\underbrace {\int \_{-\infty }^{\infty }x\_{2}(t+{\tfrac {\tau }{2}})x\_{1}^{\*}(t-{\tfrac {\tau }{2}})e^{-j2\pi t\eta }\,dt} \_{A\_{{x\_{2}}{x\_{1}}}(\tau ,\eta )}}
其中 {\displaystyle {\begin{cases}Auto-terms:\quad A\_{x\_{1}}(\tau ,\eta ),\ A\_{x\_{2}}(\tau ,\eta )\\Cross-terms:\ A\_{{x\_{1}}{x\_{2}}}(\tau ,\eta ),\ A\_{{x\_{2}}{x\_{1}}}(\tau ,\eta )\end{cases}}}

#### Auto - terms

{\displaystyle A\_{x\_{1}}\left(\tau ,\eta \right)={\sqrt {\tfrac {1}{2\alpha \_{1}}}}\ e^{-\pi ({\tfrac {\alpha \_{1}\tau ^{2}}{2}}+{\tfrac {\eta ^{2}}{2\alpha \_{1}}})}\ e^{j2\pi (f\_{1}\tau -t\_{1}\eta )}}
{\displaystyle A\_{x\_{2}}\left(\tau ,\eta \right)={\sqrt {\tfrac {1}{2\alpha \_{2}}}}\ e^{-\pi ({\tfrac {\alpha \_{2}\tau ^{2}}{2}}+{\tfrac {\eta ^{2}}{2\alpha \_{2}}})}\ e^{j2\pi (f\_{2}\tau -t\_{2}\eta )}}

  

#### Cross - terms

(1) {\displaystyle \alpha \_{1}\neq \alpha \_{2}}

{\displaystyle A\_{{x\_{1}}{x\_{2}}}(\tau ,\eta )={\sqrt {\tfrac {1}{(\alpha \_{1}+\alpha \_{2})}}}\ e^{-\pi ({\tfrac {(\alpha \_{1}+\alpha \_{2})(\tau -t\_{1}+t\_{2})^{2}}{4}}\ +\ {\tfrac {[(\alpha \_{1}-\alpha \_{2})(\tau -t\_{1}+t\_{2})-j2(\eta -f\_{1}+f\_{2})]^{2}}{4(\alpha \_{1}+\alpha \_{2})}})}\ e^{j2\pi [({\tfrac {f\_{1}+f\_{2}}{2}})\tau -{\tfrac {t\_{1}+t\_{2}}{2}}\eta +{\tfrac {(f\_{1}-f\_{2})(t\_{1}+t\_{2})}{2}}]}}
{\displaystyle ={\sqrt {\tfrac {1}{2\alpha \_{u}}}}\ e^{-\pi ({\tfrac {\alpha \_{u}(\tau -t\_{d})^{2}}{2}}\ +\ {\tfrac {[\alpha \_{d}(\tau -t\_{d})-j2(\eta -f\_{d})]^{2}}{8\alpha \_{u}}})}\ e^{j2\pi (f\_{u}\tau -t\_{n}\eta +f\_{d}t\_{u})}}

{\displaystyle A\_{{x\_{2}}{x\_{1}}}(\tau ,\eta )=A\_{{x\_{1}}{x\_{2}}}^{\*}(-\tau ,-\eta )}
{\displaystyle {\begin{cases}t\_{u}={\tfrac {t\_{1}+t\_{2}}{2}},\ f\_{u}={\tfrac {f\_{1}+f\_{2}}{2}},\ \alpha \_{u}={\tfrac {\alpha \_{1}+\alpha \_{2}}{2}}\\t\_{d}=t\_{1}-t\_{2},\ f\_{d}=f\_{1}-f\_{2},\ \alpha \_{d}=\alpha \_{1}-\alpha \_{2}\end{cases}}}
(2) {\displaystyle \alpha \_{1}=\alpha \_{2}}

{\displaystyle A\_{{x\_{1}}{x\_{2}}}(\tau ,\eta )={\sqrt {\tfrac {1}{2\alpha \_{u}}}}\ e^{-\pi ({\tfrac {\alpha \_{u}(\tau -t\_{d})^{2}}{2}}\ +\ {\tfrac {(\eta -f\_{d})^{2}}{2\alpha \_{u}}})}\ e^{j2\pi (f\_{u}\tau -t\_{n}\eta +f\_{d}t\_{u})}}
{\displaystyle A\_{{x\_{2}}{x\_{1}}}(\tau ,\eta )=A\_{{x\_{1}}{x\_{2}}}^{\*}(-\tau ,-\eta )}
因此，我們目前得到 {\displaystyle A\_{x\_{1}}\left(\tau ,\eta \right),A\_{x\_{2}}\left(\tau ,\eta \right)} (auto-terms) 和 {\displaystyle A\_{{x\_{1}}{x\_{2}}}(\tau ,\eta ),A\_{{x\_{2}}{x\_{1}}}(\tau ,\eta )} (cross-terms) 的公式，我們再仔細的分析 auto-terms 和 cross-terms 分別發生最大值的位置。

 Ambiguity Function 分析圖 
首先，先看 Auto-terms：

{\displaystyle |A\_{x\_{1}}\left(\tau ,\eta \right)|}最大值發生在 {\displaystyle \tau =0,\eta =0}的地方
{\displaystyle |A\_{x\_{2}}\left(\tau ,\eta \right)|}最大值發生在 {\displaystyle \tau =0,\eta =0}的地方
而 Cross-terms：

{\displaystyle |A\_{{x\_{1}}{x\_{2}}}(\tau ,\eta )|}最大值發生在 {\displaystyle \tau =t\_{d},\eta =f\_{d}}的地方
{\displaystyle |A\_{{x\_{2}}{x\_{1}}}(\tau ,\eta )|}最大值發生在 {\displaystyle \tau =-t\_{d},\eta =-f\_{d}}的地方
換句話說，如果我們繪製一個 x軸為 {\displaystyle \tau }, y軸為 {\displaystyle \eta } 的座標圖，Auto-terms發生在原點 {\displaystyle (0,0)} 的位置，而 Cross-terms 則是以原點為對稱中心，在第一象限和第三象限的位置，

這也是為什麼可以透過一個低通函數來濾除雜訊，把主成分 Auto-terms 分離出來，避免交叉項的問題。

與 維格納分布 Wigner Distribution Function 的不同
----------------------------------------

維格納分布是由尤金·維格納於 1932 年提出的新的時頻分析方法，對於非穩態的訊號有不錯的表現。

相較於傅立葉轉換或是短時距傅立葉轉換，維格納分布能有比較好的解析能力。

維格納分布的定義為:

{\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 }
如果我們假設 x(t) 是一個具有弦波特性的訊號, {\displaystyle x(t)=e^{j2\pi f\_{0}t}}

那麼將此 x(t) 代入維格納分布中， 

 Wigner Distribution Function 分析圖
{\displaystyle W\_{x}(t,f)=\int \_{-\infty }^{\infty }e^{j2\pi f\_{0}(t+{\tfrac {\tau }{2}})}e^{-j2\pi f\_{0}(t-{\tfrac {\tau }{2}})}\ e^{-j2\pi \tau f}\ d\tau }
{\displaystyle =\int \_{-\infty }^{\infty }e^{j2\pi f\_{0}\tau }\ e^{-j2\pi \tau f}\ d\tau }
{\displaystyle =\int \_{-\infty }^{\infty }e^{-j2\pi \tau (f-f\_{0})}d\tau }
{\displaystyle =\delta (f-f\_{0})}
所以當 {\displaystyle x(t)=e^{j2\pi f\_{0}t}} 時，{\displaystyle W\_{x}(t,f)} 在 {\displaystyle f=f\_{0}} 的地方會有最大值。

換句話說，當 {\displaystyle x(t)}有 modulation {\displaystyle f\_{0}} 或是有 time shifting t\_{0} 的情況發生時，會影響維格納分布 (Wigner Distribution Function) 最大值 {\displaystyle |W\_{x}(t,f)|} 的位置

然而，對於科恩系列分布 (Cohen's class distribution)而言，time shifting t\_{0} 和 modulation f\_0 並不會影響 {\displaystyle |A\_{x}\left(\tau ,\eta \right)|}

參考
--

* Jian-Jiun Ding, Time frequency analysis and wavelet transform class note, the Department of Electrical Engineering, National Taiwan University (NTU), Taipei, Taiwan, 2007.
* Jian-Jiun Ding, Time frequency analysis and wavelet transform class note, the Department of Electrical Engineering, National Taiwan University (NTU), Taipei, Taiwan, 2018.