# 科恩克萊斯分佈

## 數學定義

$C_x(t, f)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}A_x(\eta,\tau)\Phi(\eta,\tau)\exp (j2\pi(\eta t-\tau f))\, d\eta\, d\tau,$

## Cohen's class分佈系列函數

File:WDF.jpg

### 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

$C_x(t, f)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}A_x(\eta,\tau)\Phi(\eta,\tau)\exp (j2\pi(\eta t-\tau f))\, d\eta\, d\tau,$
$=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}x(u+\frac{\tau}{2})x^*(u-\frac{\tau}{2})\Phi(\eta,\tau)e^{-j2\pi u\eta+j2\pi(\eta t-\tau f)}dud\tau d\eta$

### 簡化方法一:不是所有的$A_x(\eta,\tau)$的值都要計算出

$\ \left|\eta\right|>B\$$\ \left|\tau\right|>C$，若$\Phi(\eta,\tau)=0$，則$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

$C_x(t, f)=\int_{-C}^{C}\int_{-\infty}^{\infty}x(u+\frac{\tau}{2})x^*(u-\frac{\tau}{2})[\int_{-B}^{B}\Phi(\eta,\tau)e^{-j2\pi,\eta(t-u)}d\eta]e^{-j2\pi\tau,f}dud\tau$
$=\int_{-C}^{C}\int_{-\infty}^{\infty}x(u+\frac{\tau}{2})x^*(u-\frac{\tau}{2})\Phi(\tau,t-u)e^{-j2\pi\tau,f}dud\tau$，其中
$\Phi(\tau,t-u)=\int_{-B}^{B}\Phi(\eta,\tau)e^{-j2\pi,\eta(t-u)}d\eta$，由於$\Phi(\tau,t-u)$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)

$C_x(t, f)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}x(u+\frac{\tau}{2})x^*(u-\frac{\tau}{2})\phi(t-u,\tau)due^{-j2\pi f\tau}d\tau$，其中
$\phi(t,\tau)=\int_{-\infty}^{\infty}\Phi(\eta,\tau)exp(j2\pi\eta t)d\eta$。對$\left|t\right|>b$或是$\left|\tau\right|>c$，則
$C_x(t, f)=\int_{-c}^{c}\int_{t-b}^{t+b}x(u+\frac{\tau}{2})x^*(u-\frac{\tau}{2})\phi(t-u,\tau)due^{-j2\pi f\tau}d\tau$，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.