# 改進型韋格納分佈

## 原理

### 韋格納分佈的數學定義

${\displaystyle W_{x}(t,f)=\int _{-\infty }^{\infty }x(t+\tau /2)x^{*}(t-\tau /2)e^{-j2\pi \tau \,f}d\tau }$${\displaystyle =\int _{-\infty }^{\infty }X(f+\eta /2)\cdot X^{*}(f-\eta /2)e^{j2\pi t\eta }\cdot d\eta }$

### 改進型韋格納分佈的數學定義

• 定義一：:${\displaystyle W_{x}(t,f)=\int _{-\infty }^{\infty }w(\tau /2)w^{*}(-\tau /2)x(t+\tau /2)x^{*}(t-\tau /2)e^{-j2\pi \tau \,f}d\tau }$
，其中w(t)為遮罩函數. 常為方波，其方波寬度為參數B。可寫成 ${\displaystyle w(t)={\begin{cases}1\ \ \ if\ |t|
• 定義二：:${\displaystyle W_{x}(t,f)=\int _{-\infty }^{\infty }P(\theta )Y(t,f+\theta /2)Y^{*}(t,f-\theta /2)d\theta }$
， 其中 ${\displaystyle Y(t,f)=\int _{-\infty }^{\infty }w(\tau )x(t+\tau )e^{-j2\pi f\tau }d\tau }$； ${\displaystyle P(\theta )\,}$ 類似遮罩函數，
${\displaystyle P(\theta )\approx 1\ }$， 當θ很小
${\displaystyle P(\theta )\approx 0\ }$， 當θ很大

• 定義三：${\displaystyle W_{x}(t,f)=\int _{-\infty }^{\infty }w(\tau )x^{L}(t+{\tfrac {\tau }{2L}})\cdot {\overline {x^{L}(t-{\tfrac {\tau }{2L}})}}e^{-j2\pi \tau f}\cdot d\tau }$

• 定義四：${\displaystyle W_{x}(t,f)=\int _{-\infty }^{\infty }[\prod _{l=1}^{q/2}x(t+d_{l}\tau )x^{*}(t-d_{-l}\tau )]e^{-j2\pi \tau f}d\tau }$

## 性能表現

1. ${\displaystyle x(t)={\begin{cases}cos(3\pi t)\ \ \ t\leq -4\\cos(6\pi t)\ \ \ -44\end{cases}}}$

1. ${\displaystyle x(t)=exp(j\cdot (t-5)^{3}-j\cdot 6\pi \cdot t)}$

## 參考資料

• Jian-Jiun Ding, class lecture of Time Frequency Analysis and Wavelet transform, Graduate Institute of Communication Engineering, National Taiwan University, Taipei, Taiwan, 2007.
• L. J. Stankovic, S. Stankovic, and E. Fakultet, 「An analysis of instantaneous frequency representation using time frequency distributions-generalized Wigner distribution,」 IEEE Trans. on Signal Processing, pp. 549-552, vol. 43, no. 2, Feb. 1995
• Jian-Jiun Ding, Time frequency analysis and wavelet transform class notes, Graduate Institute of Communication Engineering, National Taiwan University (NTU), Taipei, Taiwan, 2017.