改進型韋格納分佈

${\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 }$

為了改善韋格納分佈的相交項(cross-term)問題，改進型韋格納分佈在此引入了一個類似遮罩(mask)的函數，將相交項過濾掉。

• • 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 ${\displaystyle w(t)={\begin{cases}1\ \ \ if\ |t|<B\\0\ \ \ otherwise\end{cases}}}$

• • 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

適當地選擇${\displaystyle P(\theta )\approx 1\ }$的範圍。若選的範圍太小，將會破壞原本的項(auto term)。

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

增加 L 可以減少相交項(cross-term)的影響(但是不會完全消除)

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

當 q = 2 和 ${\displaystyle d_{l}=d_{-l}=0.5}$，就是原本的韋格納分佈。

當指數函數的次項不超過 q/2 +1時，就可以避免相交項(cross-term)

然而，相交項(cross-term)會介於兩個訊號之間，無法完全被移除。

<說明>

定義四的維格納分布又稱為多項型維格納分布 (Polynomial Wigner Distribution Function)

${\displaystyle W_{x}(t,f)=\int _{-\infty }^{\infty }e^{j2\pi \sum _{n=1}^{{\tfrac {q}{2}}+1}na_{n}t^{n-1}\tau }\ e^{-j2\pi \tau f}d\tau =\int _{-\infty }^{\infty }[\prod _{\ell =1}^{q/2}x(t+d_{\ell }\tau )x^{*}(t-d_{-\ell }\tau )]e^{-j2\pi \tau f}d\tau }$

If ${\displaystyle x(t)=e^{j2\pi \sum _{n=1}^{{\tfrac {q}{2}}+1}na_{n}t^{n}}}$

所以 ${\displaystyle d_{\ell }}$ 必須要能滿足下面的式子：

${\displaystyle e^{j2\pi \sum _{n=1}^{{\tfrac {q}{2}}+1}na_{n}t^{n-1}\tau }=\prod _{\ell =1}^{q/2}x(t+d_{\ell }\tau )x^{*}(t-d_{-\ell }\tau )}$

${\displaystyle W_{x}(t,f)=\int _{-\infty }^{\infty }e^{-j2\pi (f-\sum _{n=1}^{{\tfrac {q}{2}}+1})na_{n}t^{n-1}\tau }d\tau \cong \delta (f-\sum _{n=1}^{{\tfrac {q}{2}}+1}na_{n}t^{n-1}\tau )}$

其中 ${\displaystyle \sum _{n=1}^{{\tfrac {q}{2}}+1}na_{n}t^{n-1}}$為 ${\displaystyle x(t)}$ 的瞬時頻率

接下來，我們來看 ${\displaystyle d_{\ell }}$ 要怎麼設定：

(1) 當 ${\displaystyle q=2}$ 的時候： ${\displaystyle x(t+d_{\ell }\tau )x^{*}(t-d_{-\ell }\tau )=e^{j2\pi \sum _{n=1}^{2}na_{n}t^{n-1}\tau }}$

如果我們把 ${\displaystyle x(t)=e^{j2\pi \sum _{n=1}^{{\tfrac {q}{2}}+1}a_{n}t^{n}}}$代入，可以得到下列式子：

${\displaystyle a_{2}(t+d_{1}\tau )^{2}+a_{1}(t+d_{1}\tau )-a_{2}(t-d_{-1}\tau )-a_{1}(t-d_{-1}\tau )=2a_{2}t\tau +a_{1}\tau }$

${\displaystyle {\begin{cases}d_{1}+d_{-1}=1\\d_{1}-d_{-1}=0\end{cases}}}$ ${\displaystyle \Longrightarrow {\begin{cases}d_{1}={\tfrac {1}{2}}\\d_{-1}={\tfrac {1}{2}}\end{cases}}}$

由此可以知道，當 ${\displaystyle q=2}$ 並且 ${\displaystyle d_{-1}=d_{1}={\tfrac {1}{2}}}$ 時，多項型的維格納分布 (Polynomial Wigner Distribution Function) 就會與原始的維格納分布相同

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

(2) 當 ${\displaystyle q=4}$ 的時候： ${\displaystyle x(t+d_{\ell }\tau )x^{*}(t-d_{-\ell }\tau )=e^{j2\pi \sum _{n=1}^{3}na_{n}t^{n-1}\tau }}$

如果我們把 ${\displaystyle x(t)=e^{j2\pi \sum _{n=1}^{{\tfrac {q}{2}}+1}a_{n}t^{n}}}$代入，可以得到下列式子：

${\displaystyle a_{3}(t+d_{1}\tau )^{3}+a_{2}(t+d_{1}\tau )^{2}+a_{1}(t+d_{1}\tau )a_{3}(t+d_{2}\tau )^{3}+a_{2}(t+d_{2}\tau )^{2}+a_{1}(t+d_{2}\tau )-a_{3}(t+d_{-1}\tau )^{3}-a_{2}(t+d_{-1}\tau )^{2}-a_{1}(t+d_{-1}\tau )-a_{3}(t+d_{-2}\tau )^{3}-a_{2}(t+d_{-2}\tau )^{2}-a_{1}(t+d_{-2}\tau )}$

${\displaystyle =3a_{3}t^{2}\tau +2a_{2}t\tau +a_{1}\tau }$

${\displaystyle {\begin{cases}a_{3}(t+d_{1}\tau )^{3}+a_{3}(t+d_{2}\tau )^{3}-a_{3}(t+d_{-1}\tau )^{3}-a_{3}(t+d_{-2}\tau )^{3}\\a_{2}(t+d_{1}\tau )^{2}+a_{2}(t+d_{2}\tau )^{2}-a_{2}(t+d_{-1}\tau )^{2}-a_{2}(t+d_{-2}\tau )^{2}\\a_{1}(t+d_{1}\tau )+a_{1}(t+d_{2}\tau )-a_{1}(t+d_{-1}\tau )-a_{1}(t+d_{-2}\tau )\end{cases}}}$${\displaystyle ={\begin{cases}3a_{3}t^{2}\tau \\2a_{2}t\tau \\a_{1}\tau \end{cases}}}$

所以我們可以得到

${\displaystyle {\begin{cases}d_{1}+d_{2}+d_{-1}+d_{-2}=1\\{d_{1}}^{2}+{d_{2}}^{2}-{d_{-1}}^{2}-{d_{-2}}^{2}=0\\{d_{1}}^{3}+{d_{2}}^{3}+{d_{-1}}^{3}+{d_{-2}}^{3}=0\end{cases}}}$

可以看到如果 ${\displaystyle q}$ 太大，${\displaystyle d_{\ell }}$ 會不好設計。