# 科恩克萊斯分佈

## 數學定義

${\displaystyle C_{x}(t,f)=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }A_{x}(\eta ,\tau )\Phi (\eta ,\tau )\exp(j2\pi (\eta t-\tau f))\,d\eta \,d\tau ,}$

## Cohen's class分佈系列函數

### 錐狀分布(Cone-Shape Distribution)

File:400px-Choi williams.jpg

### 喬伊-威廉斯(Choi-Williams)

File:400px-Cone shape 2.jpg

## Cohen's class分佈優缺點

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。

1. 需要較高的計算量與時間。
2. 缺乏良好的數學特性。

## Cohen's class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

${\displaystyle \ \left|\eta \right|>B\ }$${\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，${\displaystyle \eta }$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。

## 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.