# 時頻分析中的核方法

## 時頻分布的形式

Choi-Williams分布 ${\displaystyle e^{\frac {-\theta ^{2}\tau ^{2}}{\sigma }}}$
Page分布 ${\displaystyle e^{j\theta |\tau |}}$

## 時頻分布的不同形式

### 特徵函數

${\displaystyle M(\theta \,,\tau )=\int \int s^{*}(u-{\frac {1}{2}}\tau )s(u+{\frac {1}{2}}\tau )\phi (\theta \,,\tau )e^{j\theta u}du=\phi (\theta \,,\tau )A(\theta \,,\tau )}$

${\displaystyle C(t,\omega )={\frac {1}{4\pi ^{2}}}\int \int M(\theta \,,\tau )e^{-j\theta t-j\tau \omega }d\theta d\tau }$

### 自相關函數

${\displaystyle R_{t}(\tau )={\frac {1}{2\pi }}\int \int s^{*}(u-{\frac {1}{2}}\tau )s(u+{\frac {1}{2}}\tau )\phi (\theta \,,\tau )e^{j\theta (u-t)}d\theta du}$

${\displaystyle C(t,\omega )={\frac {1}{2\pi }}\int R_{t}(\tau )e^{-j\omega \tau }d\tau }$

### 傅立葉變換

${\displaystyle r(t,\tau )={\frac {1}{2\pi }}\int \phi (\theta ,\tau )e^{-jt\theta }d\theta }$
${\displaystyle C(t,\omega )={\frac {1}{2\pi }}\int \int r(t-u,\tau )s^{*}(u-{\frac {1}{2}}\tau )s(u+{\frac {1}{2}}\tau )e^{-j\omega \tau }d\tau du}$

### 雙線性變換

${\displaystyle C(t,\omega )=\int \int K(t,\omega ,x',x)s^{*}(x')s(x)dx'dx}$
${\displaystyle K(t,\omega ,x',x)={\frac {1}{2\pi }}r(t-{\frac {1}{2}}(x+x'),x'-x)e^{-j\omega (x'-x)}}$

## 時頻分布的性質與核

### 邊際條件

${\displaystyle \int C(t,\omega )d\omega ={\frac {1}{2\pi }}\int \int \phi (\theta ,0)|s(u)|^{2}e^{j\theta (u-t)}=|s(u)|^{2}}$

### 實數性

${\displaystyle C(t,\omega )={\frac {1}{4\pi ^{2}}}\int \int M(\theta \,,\tau )e^{-j\theta t-j\tau \omega }d\theta d\tau }$

${\displaystyle M(\theta \,,\tau )=M^{*}(-\theta \,,-\tau )}$

${\displaystyle \phi (\theta \,,\tau )=\phi ^{*}(-\theta \,,-\tau )}$

### 逆變換

${\displaystyle s^{*}(u-{\frac {1}{2}}\tau )s(u+{\frac {1}{2}}\tau )={\frac {1}{2\pi }}\int {\frac {M(\theta \,,\tau )}{\phi (\theta \,,\tau )}}e^{-j\theta u}d\theta }$

${\displaystyle s(t)={\frac {1}{2\pi s^{*}(0)}}\int {\frac {M(\theta \,,\tau )}{\phi (\theta \,,\tau )}}e^{-j\theta u}d\theta ={\frac {1}{2\pi s^{*}(0)}}\int {\frac {C(t',\omega )}{\phi (\theta \,,\tau )}}e^{jt\omega +j\theta (t'-{\frac {t}{2}})}dt'd\omega d\theta }$

## 不同時頻分布之間的關係

${\displaystyle C_{1}(t,\omega )={\frac {1}{4\pi ^{2}}}\int \int \int \int {\frac {\phi _{1}(\theta \,,\tau )}{\phi _{2}(\theta \,,\tau )}}C_{2}(t',\omega ')e^{j\theta (t'-t)+j\tau (\omega -\omega ')}}$

## 參考資料

• Leon Cohen, "Generalized phase-space distribution functions," Jour. Math. Phys., vol. 7, pp. 781-786, 1966.
• Leon Cohen, "Time-frequency analysis," 1995.