# 最小相位

## 逆系統

${\displaystyle \mathbb {H} _{inv}\,\mathbb {H} =\mathbb {I} }$

${\displaystyle \mathbb {H} \,{\tilde {x}}={\tilde {y}}}$

${\displaystyle {\tilde {y}}}$ 作為逆系統的輸入，可得：

${\displaystyle \mathbb {H} _{inv}\,{\tilde {y}}=\mathbb {H} _{inv}\,\mathbb {H} \,{\tilde {x}}=\mathbb {I} \,{\tilde {x}}={\tilde {x}}}$

### 離散時間的例子

${\displaystyle (h*h_{inv})(n)=\sum _{k=-\infty }^{\infty }h(k)\,h_{inv}(n-k)=\delta (n)}$

## 最小相位系統

### 因果性

${\displaystyle h(n)=0\,\,\forall \,n<0}$

${\displaystyle h_{inv}(n)=0\,\,\forall \,n<0}$

### 穩定性

${\displaystyle \sum _{n=-\infty }^{\infty }{\left|h(n)\right|}=\|h\|_{1}<\infty }$

${\displaystyle \sum _{n=-\infty }^{\infty }{\left|h_{inv}(n)\right|}=\|h_{inv}\|_{1}<\infty }$

## 頻域分析

### 離散時間系統的頻域分析

${\displaystyle (h*h_{inv})(n)=\,\!\delta (n)}$

${\displaystyle H(z)\,H_{inv}(z)=1}$

${\displaystyle H_{inv}(z)={\frac {1}{H(z)}}}$

${\displaystyle H(z)={\frac {A(z)}{D(z)}}}$

${\displaystyle H_{inv}(z)={\frac {D(z)}{A(z)}}}$

### 連續時間系統的頻域分析

${\displaystyle (h*h_{inv})(t)=\,\!\delta (t)}$

${\displaystyle \delta (t)*x(t)=\int _{-\infty }^{\infty }\delta (t-\tau )x(\tau )d\tau =x(t)}$

${\displaystyle H(s)\,H_{inv}(s)=1}$

${\displaystyle H_{inv}(s)={\frac {1}{H(s)}}}$

${\displaystyle H(s)={\frac {A(s)}{D(s)}}}$

${\displaystyle H_{inv}(s)={\frac {D(s)}{A(s)}}}$

${\displaystyle H(s)}$ 的因果性及穩定性表示A (s)的所有零點都在左半S平面內，因此最小相位系統的最有極點及零點都需要嚴格的在左半S平面內。

### 增益響應及相位響應的關係

${\displaystyle H(j\omega )\ {\stackrel {\mathrm {def} }{=}}\ H(s){\Big |}_{s=j\omega }\ }$

${\displaystyle \arg \left[H(j\omega )\right]=-{\mathcal {H}}\lbrace \log \left(|H(j\omega )|\right)\rbrace \ }$

${\displaystyle \log \left(|H(j\omega )|\right)=\log \left(|H(j\infty )|\right)+{\mathcal {H}}\lbrace \arg \left[H(j\omega )\right]\rbrace \ }$ .

${\displaystyle H(j\omega )=|H(j\omega )|e^{j\arg \left[H(j\omega )\right]}\ {\stackrel {\mathrm {def} }{=}}\ e^{\alpha (\omega )}e^{j\phi (\omega )}=e^{\alpha (\omega )+j\phi (\omega )}\ }$

${\displaystyle \phi (\omega )=-{\mathcal {H}}\lbrace \alpha (\omega )\rbrace \ }$

${\displaystyle \alpha (\omega )=\alpha (\infty )+{\mathcal {H}}\lbrace \phi (\omega )\rbrace \ }$ .

${\displaystyle {\mathcal {H}}\lbrace x(t)\rbrace \ {\stackrel {\mathrm {def} }{=}}\ {\widehat {x}}(t)={\frac {1}{\pi }}\int _{-\infty }^{\infty }{\frac {x(\tau )}{t-\tau }}\,d\tau \ }$  .

## 時域下的最小相位

${\displaystyle \sum _{n=m}^{\infty }\left|h(n)\right|^{2}\,\,\,\,\,\,\,\forall \,m\in \mathbb {Z} ^{+}}$

## 最小相位及最小群延遲

${\displaystyle a=\left|a\right|e^{i\theta _{a}}\,{\mbox{ where }}\,\theta _{a}={\mbox{Arg}}(a)}$

${\displaystyle \phi _{a}\left(\omega \right)={\mbox{Arg}}\left(1-ae^{-i\omega }\right)}$
${\displaystyle ={\mbox{Arg}}\left(1-\left|a\right|e^{i\theta _{a}}e^{-i\omega }\right)}$
${\displaystyle ={\mbox{Arg}}\left(1-\left|a\right|e^{-i(\omega -\theta _{a})}\right)}$
${\displaystyle ={\mbox{Arg}}\left(\left\{1-\left|a\right|cos(\omega -\theta _{a})\right\}+i\left\{\left|a\right|sin(\omega -\theta _{a})\right\}\right)}$
${\displaystyle ={\mbox{Arg}}\left(\left\{\left|a\right|^{-1}-\cos(\omega -\theta _{a})\right\}+i\left\{\sin(\omega -\theta _{a})\right\}\right)}$

${\displaystyle \phi _{a}(\omega )}$ 所貢獻的相延遲如下。

${\displaystyle -{\frac {d\phi _{a}(\omega )}{d\omega }}={\frac {\sin ^{2}(\omega -\theta _{a})+\cos ^{2}(\omega -\theta _{a})-\left|a\right|^{-1}\cos(\omega -\theta _{a})}{\sin ^{2}(\omega -\theta _{a})+\cos ^{2}(\omega -\theta _{a})+\left|a\right|^{-2}-2\left|a\right|^{-1}\cos(\omega -\theta _{a})}}}$
${\displaystyle -{\frac {d\phi _{a}(\omega )}{d\omega }}={\frac {\left|a\right|-\cos(\omega -\theta _{a})}{\left|a\right|+\left|a\right|^{-1}-2\cos(\omega -\theta _{a})}}}$

${\displaystyle {\mbox{Arg}}\left(\prod _{i=1}^{N}\left(1-a_{i}z^{-1}\right)\right)=\sum _{i=1}^{N}{\mbox{Arg}}\left(1-a_{i}z^{-1}\right)}$

## 非最小相位系統

### 最大相位系統

• 離散時間系統下的零點都在單位圓外。
• 連續時間系統下的零點都在複數平面的右半邊。

${\displaystyle {\frac {s+10}{s+5}}\qquad {\text{and}}\qquad {\frac {s-10}{s+5}}}$

### 混合相位系統

${\displaystyle {\frac {(s+1)(s-5)(s+10)}{(s+2)(s+4)(s+6)}}}$

## 參考資料

1. ^ Hassibi, Babak; Kailath, Thomas; Sayed, Ali H. Linear estimation. Englewood Cliffs, N.J: Prentice Hall. 2000: 193. ISBN 0-13-022464-2.
2. ^ J. O. Smith III, Introduction to Digital Filters with Audio Applications (September 2007 Edition).

## 延伸閱讀

• Dimitris G. Manolakis, Vinay K. Ingle, Stephen M. Kogon : Statistical and Adaptive Signal Processing, pp. 54–56, McGraw-Hill, ISBN 0-07-040051-2
• Boaz Porat : A Course in Digital Signal Processing, pp. 261–263, John Wiley and Sons, ISBN 0-471-14961-6